If x 1 , x 2 , … , x n are in H.P, then ∑ r = 1 n − 1 x r x r + 1 is equal to
The sum to n terms of the series 1 + 2 1 + 1 n + 3 1 + 1 n 2 + … . . . is given by
The sum of first two terms of Geometric Progression is 12 and the sum of third and fourth terms is 48. If the terms of Geometric Progression are alternatively positive and negative, then first term is
The ratio of sum of first three terms of a G.P to the sum of first six terms is 64:91 , the common ratio of G.P. is
Value of y = ( 0.36 ) log 1 4 1 3 + 1 3 2 + 1 3 3 + … upto ∞
The sum of an infinite number of terms of a G.P. is 20, and the sum of their squares is 100, then the first term of the G.P is
Product of 8 GM’s inserted between 2 and 3
Of 6 GM’s are inserted between 2 and 12 3 rd GM
The value of common ratio for which three successive terms of a GP are the sides of a triangle is
If x , y , z ,are in GP and a x = b y = c z , then
In a GP, first term is1. If 4 T 2 + 5 T 3 is minimum, then its common ratio is
Let S n denotes the sum of n terms of an A.P. whose first term is a. If the common difference d = S n – k S n – 1 + S n – 2 then k =
If the first, second and the last terms of an A.P. are a , b , c respectively, then the sum of the A . P . is
Let f : R R be such that for all x ∈ R , 2 1 + x + 2 1 – x , f ( x ) and 3 x + 3 – x are in A.P. Then the minimum value of f ( x ) is:
A G.P consisting of terms with positive terms. If each term is equal to the sum of the next two terms, then the common ratio of G.P is
If 1 1.2.3.4 + 1 2.3.4.5 + 1 3.4.5.6 + … … upto n terms = 1 18 − 1 3 f ( n ) then f ( 1 ) − f ( 3 ) is
In a triangle ABC , A , B , C are in G.P with common ratio 2 then 1 b + 1 c − 1 a is =
Let S 1 , S 2 , S 3 … and t 1 , t 2 , t 3 … are two Arithmatic Sequence such that S 1 = t 1 ≠ 0 , S 2 = 2 t 2 and ∑ i = 1 10 S i = ∑ i = 1 15 t i then S 2 − S 1 t 2 − t 1 = d 1 d 2 then d 1 − d 2 =
If 10 9 + 2 ( 11 ) 1 10 8 + 3 ( 11 ) 2 ( 10 ) 7 + . . + 10 ( 11 ) 9 = k ⋅ 10 9 then k is
Let p = 1 4 + 2 4 + 3 4 + 4 4 + … ∞ and q = 1 4 + 3 4 + 5 4 + 7 4 + … ∞ then p q is
Five numbers are in A.P whose sum is 25 and product is 2520 if one of these five numbers is – 1 2 then the greatest number amongst them is
Let a 1 , a 2 , … , a 30 be an A.P., S = ∑ i = 1 30 a i and T = ∑ i = 1 15 a ( 2 i − 1 ) . If a 5 = 27 and S − 2 T = 75 then a 10 is
If ( 1 + x ) 1 + x 2 1 + x 4 … 1 + x 128 = ∑ r = 0 n x r then n is
If x 1 , x 2 , … x n and 1 h 1 , 1 h 2 , . 1 h n are two A.P’s such that x 3 = h 2 = 8 and x 8 = h 7 = 20 then x 5 h 10 is
The value of S = 1 + 1 1 2 + 1 2 2 + 1 + 1 2 2 + 1 3 2 + . . + 1 + 1 ( 2014 ) 2 + 1 ( 2015 ) 2 is
If the sum of an infinite decreasing G.P is 3 and the sum of the squares of its terms is 9 2 then the sum of the cubes of the terms is
If S n = 1 + 1 3 + 1 3 2 + . . + 1 3 n − 1 , n ∈ N , then the least value of ‘ n ‘ such that 3 − 2 S n < 1 100 is
If a,b,c are positive numbers in A.P such that their product is 64 then the minimum value of ‘b’ is
There are n-AM’s between 1 and 31 such that 7 th mean:(n-1) th mean = 5 : 9 then find the value of n
If sum of m A.M.’s between two positive numbers is α and sum of n A.M.’s between the same numbers is β then α β =
The sum 1 + 3 + 7 + 15 + 31 + … to 100 terms is
The value of 0.2 log 5 1 4 + 1 8 + 1 16 + …… is
Let a = 1 1 1 ….1 (55 digits), b=1 + 10 2 + …. + 10 4 , c = 1 + 10 5 + 10 10 + 10 15 + …. + 10 50 then
A man saves Rs. 200 in each of the first 3 months of his service. In each of the subsequent months his saving increases by Rs.40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after.
If a , b , c are the sides of a triangle then a b + c − a + b c + a − b + c a + b + c
How many terms are to be added to make the sum 52 in the series − 8 + − 6 + − 4 + … ?
The sum of first 24 terms of the A.P, a 1 , a 2 , a 3 , …. if it is known that a 1 + a 5 + a 10 + a 15 + a 20 + a 24 = 225
If ∑ r = 1 n T r = 3 n − 1 , then find the sum of ∑ m = 1 n 1 T m .
Sum of 10 AMs between 20 and 30
Consider an A.P a 1 , a 2 , a 3 , … … . . such that a 3 + a 5 + a 8 = 11 and a 4 + a 2 = − 2 , then the value of a 1 + a 6 + a 7 , is
Let a n be the n th term of an A.P if ∑ r = 1 100 a 2 r = α and ∑ r = 1 100 a 2 r − 1 = β then the common difference of the A.P is:
The sum to infinity of the series 1 1 + 1 1 + 2 + 1 1 + 2 + 3 + … … … … … … is equal to:
Two possible observations have arithmetic mean 3 and geometric mean 2 2 .If each observation is multiplied by 2, then harmonic mean will be
Geometric mean of 10 observations is 8. If geometric mean of first six observations is 4, then geometric mean of last four observations is
If 1 b − c , 1 c − a , 1 a − b be consecutive terms of an AP, then ( b − c ) 2 , ( c − a ) 2 , ( a − b ) 2 will be in
If the roots of the equation x 3 − 12 x 2 + 39 x − 28 = 0 are in AP, then their common difference will be
The sum of the series 5 +55 +555 +…to n terms is
If log 2 , log 2 n − 1 and 2 n + 3 are in AP, then n is equal to
Let a 1 , a 2 , a 3 , a 4 and a 5 be such that a 1 , a 2 and a 3 are in A.P , a 2 , a 3 and a 4 are in G.P.; and a 3 , a 4 and a 5 are in H.P. Then a 1 , a 3 and a 5 are in
If 2 p + 3 q + 4 r = 15 then the maximum value of p 3 q 5 r 7 is
The sum of the series 1 1 + 1 2 + 1 4 + 2 1 + 2 2 + 2 4 + 3 1 + 3 2 + 3 4 + … to n terms is
If a, b, c are respectively the xth, yth and zth terms of a G.P., then ( y − z ) log a + ( z − x ) log b + ( x − y ) log c =
The sum to n terms of the sequence log a , log ar , log ar 2 , … is
The sum of the series 1 + 2 ⋅ 2 + 3 ⋅ 2 2 + 4 ⋅ 2 3 + 5 ⋅ 2 4 + … + 100 ⋅ 2 99 is
The number of terms common to two A.P.s 3, 7, 11, …, 407 and 2, 9, 16, …, 709 is
The three numbers a, b, c between 2 and 18 are such that their sum is 25; the numbers 2, a, b are consecutive terms of an A.P. and the numbers b, c, 18 are consecutive terms of a G.P. The three numbers are
The sum of first three terms of an increasing A.P. is 9 and sum of their squares is 35 . The sum to n terms of the series is
If a , b , c are in A.P. p , q , r are in H.P. and a p , b q , c r are in G.P., then p r + r p is equal to
If the interior angles of a polygon are in A P . with common difference 5° and smallest angle is 120°, then the number of sides of the polygon is
If the sum of the series 2 , 5 , 8 , 11 , . . . is 60100 , then n , the number of terms, is
If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19……….. is (102)m, then m is equal to:
Let ∑ k = 1 10 f ( a + K ) = 16 2 10 – 1 , where the function f ( x ) satisfies f ( x + y ) = f ( x ) f ( y ) for all natural numbers x , y and f ( 1 ) = 2 , then the natural number ‘ a ‘ is
For x ∈ R , let x denote the greatest integer ≤ x , then the sum of the series − 1 3 + − 1 3 − 1 100 + − 1 3 − 2 100 + ……. + − 1 3 − 99 100 i s
A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is 1 4 . Three stones A, B and C are placed at the points (1, 1) , (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man?
The sum of the infinite series 5 13 + 55 13 2 + 555 13 3 + … . .
If a n + 1 = 2 n a n and a 1 = 1 , then log 2 a 100 is equal to
If the fourth term of an arithmetic progression is 6, the m t h term is 18 and if the A.P. has integral terms only, then the number of such A.P.s is
If 0 < θ < π , then minimum value of 3 sin θ + cosec 3 θ is
In any A.P., if the sum of first six terms is 5 times the sum of next six terms then which of its term is zero?
If three real numbers x , y , z are in G.P. such that x + y + z = 42 . If x , 5 y 4 , z are in A.P., then the largest possible value of x is
Let a 1 , a 2 , a 3 , … , a 10 be in G.P. with a 51 = 25 and ∑ i = 1 101 a i = 125 . Then the value of ∑ i = 1 101 1 a i is
If a , b , c are in geometric progression then the value of a − b b − c is
Let a 1 , a 2 , a 3 … be an A.P such that a 1 + a 2 + … + a p a 1 + a 2 + … + a q = p 3 q 3 , p ≠ q then a 6 a 21 is
The sum of the first n-terms of the series 3 + 75 + 243 + 507 + … is 435 3 then n is
If the 10 th term of an A.P is 1 20 and its 20 th term is 1 10 , then the sum of first 200 terms is
If x = 30 ∘ then the sum of the series 1 + cos 2 x + cos 2 2 x + … + ∞ is
The sum to n-terms of the series 2,5,10,17,26,… is
If the first term of A.P is 2 and the sum of first five terms is equal to one fourth of the sum of the next five terms then the sum of first 30 terms is
Let S n denote the sum of the first n-terms of an A.P. If S 2 n = 3 ⋅ S n then S 3 n ÷ S n
If ( 1 + x ) 1 + x 2 1 + x 4 … 1 + x 128 = ∑ i = 0 n x r then n is
If it is given that 1 1 4 + 1 2 4 + 1 3 4 + … ∞ is π 4 90 then the value of 1 1 4 + 1 3 4 + 1 5 4 + … ∞
Consider the A.P a 1 , a 2 , a 3 … a n , the G.P b 1 , b 2 … b n such that a 1 = b 1 = 1 : a 9 = b 9 and ∑ i = 1 9 a r = 369 then
If three successive terms of a G.P with common ratio r ( r > 1 ) , form the sides of a triangle ABC and [ r ] denotes greater integer function then | [ r ] + [ − r ] | is
If the ratio of the sum to n terms of two A.P.’s is 5 n + 3 : 3 n + 4 then the ratio of their 17th terms is
Let a 1 , a 2 , a 3 , … a 4001 are in A.P. such that 1 a 1 a 2 + 1 a 2 a 3 + … . + 1 a 4000 a 4001 = 10 and a 2 + a 4000 = 50 then a 1 − a 4001 is equal to
If there successive terms of a G.P with common ratio r(r>1) form the sides of a Δ A B C and [r] denotes greates integer function, then [ r ] + [ – r ] =
If H P H 2 , … . H 20 harmonic means between 2 and 3, then H 1 + 2 H 1 − 2 + H 20 + 3 H 20 − 3
If a , b , c , d are positive real numbers such that a + b + c + d = 2 , then M = ( a + b ) ( c + d ) satisfies the relation
Let T r be the r th term of an A.P. whose first term is a and common difference is d . If for some positive integers m , n , m ≠ n , T m = 1 n and T n = 1 m , then a-d
If the p th , q t h , and r th terms of an A.P. are in G.P. then common ratio of the G.P is
After striking the floor, a certain ball rebounds (4/5)th of height from which it has fallen. Then the total distance that it travels before coming to rest, If it is gently dropped from a height of 120 m is
If a 1 , a 2 , a 3 , … … a n are in H.P then a 1 a 2 + a 2 a 3 + a 3 a 4 + … + a n − 1 a n =
If ∑ T r = n ( n + 1 ) ( n + 2 ) 3 then lim n ∞ ∑ r = 1 n 2008 T r =
Let a 1 , a 2 … a n be the terms of an A.P. a 1 + a 2 + … a p a 1 + a 2 + … + a 1 = p 2 q 2 , p ≠ q . T h e n a 6 a 21 =
If the third term of G.P is 4, then product of first 5 terms.
If x , 2 x + 2 , and 3 x + 3 are first three terms of a G.P. then the fourth term is
Find the sum to infinity of the series 1 + 2 3 + 6 3 2 + 10 3 3 + 14 3 4 + …… is
After inserting n AM’s between 2, 38 sum of resulting profession is 200. Then n=
If 8 AMS are intersected between 2 and 3 3 rd AM
Suppose ‘a’ is a fixed real number such that a − x p x = a − y q y = a − z r z if p, q, r are in AP then x, y, z all are in
In an A.P of which a is the first , if the sum of the first p terms is zero, then the sum of the next q terms is
Let S 1 , S 2 , S 3 … … … … . and t 1 , t 2 , t 3 … … … … . are two arithmetic sequences such that S 1 = t 1 ≠ 0 ; S 2 = 2 t 2 and ∑ i = 1 10 S i = ∑ i = 1 15 t i then, the value of S 2 − S 1 t 2 − t 1 is:
If ax 3 + bx 2 + cx + d is divisible by ax 2 + c ,then a, b, c, d are in
Number of identical terms in the sequence 2, 5, 8,11, … upto 100 terms and 3, 5, 7, 9, 11, … upto 100 terms, are
If the sides of a triangle are in AP, and the greatest angle of the triangle is double the smallest angle, the ratio of the sides of the triangle is
The sum to infinity of the series 1 + 2 3 + 6 3 2 + 10 3 3 3 + 14 3 4 + … . . is
The sixth term of an AP is 2, and its common difference is greater than one. The value of the common difference of the progression so that the product of the first, fourth and fifth terms is greatest is
If a 1 , a 2 , a 3 , … , a 24 are in arithmetic progression and a 1 + a 5 + a 10 + a 15 + a 20 + a 24 = 225 then a 1 + a 2 + a 3 + … + a 23 + a 24 is equal to
if A is the arithmetic mean and G 1 , G 2 be two geometric means between any two numbers, then G 1 2 G 2 + G 2 2 G 1 is equal to
Three numbers form a GP. If the 3rd term is decreased by 64, then the three numbers thus obtained will constitute an AP. If the second term of this AP is decreased by 8, a GP will be formed again, then the numbers will be
nth term of the series 1 + 4 5 + 7 5 2 + 10 5 3 + … will be
If T n denotes the nth term of the series 2 + 3 + 6 + 11 + 18 + … then T 50 is
The sum of the series 1 ⋅ 3 ⋅ 5 + 2 ⋅ 5 ⋅ 8 + 3 ⋅ 7 ⋅ 11 + … up to n terms is
If the sum of first n terms of an AP is cn 2 , then the sum of squares of these n terms is n 4 n 2 − 1 c 2 k then k is
If S 1 is the sum of an arithmetic series of ‘n’ odd number of terms and S 2 , the sum of the terms of the series in odd places, then S 1 S 2 =
The minimum number of terms of the series 1+ 3 + 9 + 27 + … so that the sum may exceed 1000, is
i − 2 − 3 i + 4 … to 100 terms =
Number of terms in the sequence 1, 3, 6, 10, 15, …, 5050 is
There are four numbers of which the first three are in G.P. whose common ratio is 1 2 and the last three are in A.P. If the last number is two less than the first, then the four members are
If the first, second and last terms of an A.P. are a, b and 2a respectively, then its sum is
If a is the first term, d the common difference and S k the sum to k terms of an A.P., then for S kx S x to be independent of x
The sum of n terms of m A.P.s are S 1 , S 2 , S 3 , …, S m . If the first term and common difference are 1, 2, 3, …, m respectively, then S 1 + S 2 + S 3 + … + S m =
If the first, second and the last terms of an A.P. are a, b, c respectively, then the sum is
If a + 2 b + 3 c = 12 , a , b , c ∈ R + ,then ab 2 c 3 is
Let a 1 , a 2 , … , a 10 be in A.P and h 1 , h 2 . . . . . h 10 be in H.P. If a 1 = h 1 = 2 and a 10 = h 10 = 3 , then a 4 h 7 is
The sum to infinity of the series S = 1 + 2 3 + 6 3 2 + 10 3 3 + 14 3 4 + … is
If ∑ r = 1 n t r = 1 12 n ( n + 1 ) ( n + 2 ) , the value ∑ r = 1 n 1 t r is
If x = 1 + a + a 2 + … ( | a | < 1 ) and y = 1 + b + b 2 + … ( | b | < 1 ) then some of the series 1 + a b + a 2 b 2 + … is
The sum of the series S = ∑ r − 1 n log a r + 1 b r − 1 is
Sum to n terms of the series 1 1 ⋅ 2 ⋅ 3 ⋅ 4 + 1 2 ⋅ 3 ⋅ 4 ⋅ 5 + 1 3 ⋅ 4 ⋅ 5 ⋅ 6 + ⋯ is
If x denotes the greatest integer ≤ x , then value of S = [ 1 ] + [ 2 ] + [ 3 ] + … + [ 2024 ] is
If a 1 b + 1 c , b 1 c + 1 a , c 1 a + 1 b are in A.P., then
If the sum of first 2 n terms of the A.P. 2, 5, 8, …, is equal to the sum of first n terms of the A.P. 57, 59, 61,…., then n equals
If the sum of first n terms of two A.P.’s are in the ratio 3 n + 8 : 7 n + 15 , then the ratio of 12th term is
Sum of the series ∑ r = 1 n r ( r + 1 ) ! is
Let x = 2 2 2 2 … upto ∞ and y = 2 + 2 + 2 + 2 + … upto ∞ , then x – y equals
If the sum of an infinite decreasing G.P. is 3 and sum of the cubes of its terms is 108/13, then common ratio is given by
If H . M . : G . M . = 4 : 5 for the two positive numbers, then the ratio of the numbers is
The greatest positive integer k, for which 49 k + 1 is a factor of the sum 49 125 + 49 124 + …… + 49 2 + 49 + 1 , is
Five numbers are in A.P. whose sum is 25 and product is 2520, if one of these five numbers is − 1 2 , then the greatest number amongst them is
The sum ∑ k = 1 20 ( 1 + 2 + 3 + … . . + k ) is
If 5 + 9 + 13 + … upto n terms 7 + 9 + 11 + … . upto ( n + 1 ) terms = 17 16 , then n is equal to
Let a 1 , a 2 , a 3 , … . , a 49 be in A.P. such that ∑ k = 0 12 a 4 k + 1 = 416 and a 9 + a 43 = 66 . If a 1 2 + a 2 2 + … . + q 17 2 = 140 m , then m is equal to
Let a, b and c from a G.P of common ratio r, with 0 < r < 1. If a, 2b and 3c form an A.P. then r equals
If x , y , z ∈ ℝ + such that x + y + z = 4 , then maximum possible value of x y z 2 is:
Given sum of first n terms of an AP is 2n + 3n 2 . Another AP is formed with the same first term and double of the common difference, the sum of n terms of the new AP is:
i f 3 + 1 4 ( 3 + P ) + 1 4 2 ( 3 + 2 P ) + 1 4 3 ( 3 + 3 P ) + …. ∞ = 8 then the value of P is
If a 1 , a 2 , a 3 , … … … , a 20 are A.M’s between 13 and 67 , then the value of a 1 + a 2 + a 3 + … … + a 20 is equal to
The sum of the infinite number of terms of the series 1 + 4 5 + 7 5 2 + 10 5 3 + … … . is k , then the the value of 16k is
If a 1 , a 2 , a 3 , … . a 4001 are terms of an AP such that 1 a 1 a 2 + 1 a 2 a 3 + … + 1 a 4000 a 4001 = 10 and a 1 + a 4001 = 50 then a 1 − a 4001 is equal to
if a,b,c are in A.P and a + 2 b – c 4 b + 2 c – 2 a c + a – b = λ abc,then the value of λ is
The sum of 25 terms of an A.P., whose all the terms are natural numbers, lies between 1900 and 2000 and its 9 th term is 55. Then the first term of the A.P. is
Let a n be the n th term of the G.P of positive numbers. Let ∑ n = 1 100 a 2 n = α and ∑ n = 1 100 a 2 n − 1 = β such that α ≠ β , then the common ratio is
The largest term common to the sequences 1,11,21,31,… to 100 terms and 31,36,41,46,… to 100 terms is
The largest terms of the sequence 1 503 , 4 524 , 9 581 , 16 692 , … is
Let a 1 , a 2 , a 3 , … a 49 be in A.P such that ∑ k = 0 12 a 4 k + 1 = 416 and a 9 + a 43 = 66 . If a 1 2 + a 2 2 + . . + a 17 2 = 140 m then m is
Let A be the sum of the first 20 terms and ‘ B ‘ be the sum of the first 40- terms of the series 1 2 + 2.2 2 + 3 2 + 2.4 2 + 5 2 + 2.6 2 + . . If B − 2 A = 100 λ then λ is
The sum of the first 20 terms of the series 1 + 3 2 + 7 4 + 15 8 + 31 16 + … is
If a 1 , a 2 , … a n are in H.P then the expression a 1 a 2 + a 2 a 3 + . . + a n − 1 a n is
The sum of the first ten terms of the series 1 3 5 2 + 2 2 5 2 + 3 1 5 2 + 4 2 + 4 4 5 2 + . . is 16 5 m then m is
The sum of the first n terms of the series 1 2 + 2.2 2 + 3 2 + 2.4 2 + 5 2 + . Is n ( n + 1 ) 2 2 , when n is even, when ‘n’ is odd sum is
The sum of the series 2 2 + 2 ( 4 ) 2 + 3 ( 6 ) 2 + … upto 10 terms
The sum of the series 3 × 1 3 1 2 + 5 1 3 + 2 3 1 2 + 2 2 + 7 1 3 + 2 3 + 3 3 1 2 + 2 2 + 3 2 + … + upto to 10 th term is
Let S n denote the sum of first n-terms of an A.P. If S 4 = 16 , S 6 = − 48 then S 10 is
Let a 1 , a 2 , a 3 , … a 101 are in G.P with a 51 = 25 , and ∑ i = 1 101 a i = 125 . Then the value of ∑ i = 1 101 1 ai is
Let a 1 , a 2 , a 3 , … a 49 , a 50 are in A.P. If a 1 = 4 , and a 50 = 144 then the value of 1 a 1 + a 2 + 1 a 2 + a 3 + .. + 1 a 49 + a 50 =
a 1 , a 2 , a 3 … a n are in A.P such that a n = 100 , a 50 − a 49 = 3 / 5 then 15 th term of A.P from the end is
Let a,b,c form a G.P of common ratio r, with 0<r<1. If a,2b,3c forms an A.P, then common ratio r is
The 10th common term between the series 3+7+11+.. and 1+6+11+… is
The numbers 3 2 sin 2 α − 1 , 14 and 3 4 − 2 sin 2 α from first three terms of an A.P its 5 th term is
If t n = 1 4 ( n + 2 ) ( n + 3 ) for n = 1 , 2 , 3 , … then 1 t 1 + 1 t 2 + 1 t 3 + … + 1 t 2003 is
If ( 1 − p ) 1 + 2 x + 4 x 2 + 8 x 3 + 16 x 4 + 32 x 5 = 1 − p 6 ⋅ p ≠ 1 then a value of p x is
A non constant A.P has common difference d and first term is (1-ad). If the sum of the first 20 terms is 20, then the value of ‘a’ is
Consider two positive numbers a and b. If A.M. of a and b exceeds their G.M by 3/2 and G.M of a and b exceeds their H.M by 6/5 then the value of a 2 +b 2 is
The maximum sum of the series 20 + 19 1 3 + 18 2 3 + . . is
Let α , β be two distinct values of x lying in ( 0 , π ) for which 5 sin x , 10 sin x , 10 4 sin 2 x + 1 are three consecutive terms of a G.P then the minimum value of | α − β | on
The value of the sum 1.2 2 + 2.3 2 + 3.4 2 + … + 19 ⋅ 20 2 is
The sum of the infinite series 5 13 + 55 13 2 + 555 13 3 + … is
Three numbers a,b,c are between 2 and 18 such that their sum is 25 and 2,a,b are A.P and b,c,18 are in G.P then abc is
Let I n = ∫ 0 π / 4 tan n xdx , then I 2 + I 4 , I 3 + I 5 , I 4 + I 6 , I 5 + I 7 , … are in
If a n + 1 = 1 1 − a n for n ≥ 1 and a 3 = a 1 then a 2001 2001 is
The positive integer ‘ n ‘ for which 2 . 2 2 + 3 . 2 3 + 4 . 2 4 + . . + n . 2 n = 2 n + 10 is
The sum of all the integers which are divisible by ‘7’ and lying between 50 and 500 is
There are n -A.M is between 7 and 85 such that (n-3) th mean : n th mean is as 11:24 then ‘n’ is
One side of an equilateral triangle is 24 cm. The mid points of the sides are joined to form another triangle whose midpoints are joined to form still another triangle. This process continues infinitely many times, then the sum of the perimeter of all the triangles
The sum of the following series 5+55+555+… to n terms is
The sum of the first 10 -terms of the series 5 1.2.3 + 7 2.3.9 + 9 3.4.27 + … is
If the 10 th term of H.P is 21 and 21 st term of H.P is 10 then 420 th term is
The average of the numbers n sin n ∘ where n = 2 , 4 , 6 , … 180 ∘ is
If the product of three terms of a G.P is 216 and the sum of their products taken in pairs is 156 then the greatest term is
Between two numbers whose sum is 13/6 , an even number of A.M’s are inserted , the sum of these means exceeds the number of means by unity, then find the number of means
The interior angles of a convex polygon forms on A.P with common difference of 4 ∘ . If the largest interior angle is 172 ∘ then the number of sides is
Consider an A.P, a 1 , a 2 , a 3 … such that a 3 + a 5 + a 8 = 11 , and a 4 + a 2 = − 2 then the value of a 1 + a 6 + a 7 is
2 1 / 4 ⋅ 4 1 / 8 ⋅ 8 1 / 16 … ∞ is equal to
If the Harmonic mean between P and Q be H then H 1 P + 1 Q is
If log 5 2 , log 5 2 x − 3 , log 5 17 2 + 2 x – 1 are in A.P then x is
If 3 + 1 4 ( 3 + d ) + 1 4 2 ( 3 + 2 d ) + … ∞ = 8 then the value of ‘ d ‘ is
If a 1 , a 2 , a 3 … is an A.P such that a 1 + a 5 + a 10 + a 15 + a 20 + a 24 = 225 then a 1 + a 2 + a 3 + … + a 23 + a 24 is
Consider an A.P a 1 , a 2 , a 3 , … … such that a 3 + a 5 + a 8 = 11 and a 4 + a 2 = − 2 then the value of a 1 + a 6 + a 7 is
Given that n AM’s are inserted between two sets of numbers a, 2b and 2a, b where a , b ∈ R suppose further that the nth mean between these sets of numbers is same then the ratio a:b=
n A.M’s are inserted between n and 3 n + 2 then 10 th A.M. is
An employee gets Rs.300 per month in his 11th year of service and got Rs.495 per month in his 24th year of service. If his monthly salary is in A.P his initial salary and his increment are
If a 1 , a 2 , a 3 … . . a n is sequence of +Ve numbers which are in AP with common difference ‘d’ & sec a 1 sec a 2 + sec a 2 sec a 3 + … … … … . . + sec a n − 1 sec a n =
The number of common terms between the sequences given by 1, 4, 7, 10…298 and 2, 4, 6,8 …. 300 is
If the G.M of two non –zero positive numbers is to their A.M is 12:13 then numbers are in the ratio
If the 2nd , 5th and 9th terms of a non –constant A.P. are in G.P., then the common ratio of this G.P is
H 1 , H 2 are H . M ‘s between a, b then H 1 + H 2 H 1 H 2 =
If a, b, c are positive numbers such that a+b +c=1, then the minimum value of 1 a b + 1 b c + 1 c a
The sum to n terms of the series 3 1 2 + 5 1 2 + 2 2 + 7 1 2 + 2 2 + 3 2 + … is
If x = 1 + a + a 2 + … . ∞ , where | a | < 1 and y = 1 + b + b 2 + … ∞ where b <1 , then 1 + ab + 1 + a b + a 2 b 2 + … ∞ =
If log 2 ( a + b ) + log 2 ( c + d ) ≥ 4 , where a , b , c are positive numbers. Then the minimum value of expression a + b + c + d is
If a, b and c are three positive real numbers, then the minimum value of the expression b + c a + c + a b + a + b c is
The maximum sum of the series 20 + 19 1 3 + 18 2 3 + … … is
If the sum of m terms of an A.P is the same as the sum of its n terms, then the sum of its (m+n) terms is
The largest term common to the sequences 1, 11, 21, 31…….. to 100 terms and 31, 36, 41, 46……. to 100 terms is
If a r > 0 , r ∈ N and a 1 , a 2 , a 3 … . a 2 n are A.P. then a 1 + a 2 n a 1 + a 2 + a 2 + a 2 n − 1 a 2 + a 3 + a 3 + a 2 n − 2 a 3 + a 4 + … . . + a n + a n + 1 a n + a n + 1 =
If the sum of n terms of an A.P, is cn (n-1) , where c ≠ 0 , then sum of the squares of these terms is
If t n denotes nth term of the series 2 + 3 + 6 + 11 + 18 + … . then t 50
Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is
If ( p + q ) th term of a G.P. is ‘ a ‘ and its ( p − q ) th term is ‘b’ where a , b ∈ R + , then its p th term is
If a, b, c are digits, then the rational number represented by 0.Cababab…… is
If S P denotes the sum of the series 1 + r p + r 2 p + … . to ond S p denotes the sum of the series 1 − r p + r 2 p − r 3 p + … . t o ∞ , | r | < 1 , then S p + s p in terms of S 2 p is
Let S = 4 19 + 44 19 2 + 444 19 3 + … . upto ∞ . Then S is equal to
The harmonic mean of two numbers is 4. Their arithmetic mean is A and geometric mean is G. If G satisfies 2 A + G 2 = 27 then the numbers are
If the sum of the first ten terms of the series 1 3 5 2 + 2 2 5 2 + 3 1 5 2 + 4 2 + 4 4 5 2 + … . is 16 5 m , then m is equal to
The greatest value of x 2 y 3 where x > 0 , y > 0 and 3 x + 4 y = 5 is
Find the first negative term of the sequence 20 , 19 1 4 , 18 1 2 , 17 3 4 , …
In a certain A.P., 5 times the 5 th term is equal to 8 times ( 8 th term) then 13 th term is
If the sum of the series 2, 5, 8, 11, ………. n terms is 60100, then find the value of n.
Find the sum of the infinite series 1 + 4 3 + 9 3 2 + 16 3 3 + … … . . ∞
If a, b, c are in H.P and ab + bc + ca = 15, then ca=
The 5 th and 11 th terms of an H.P are 1 45 and 1 69 respectively, then find 16th term
Find the sum of the series 31 3 + 32 3 + … . + 50 3
Find the sum to n terms of the series 1 / 1 × 3 + 1 / 3 × 5 + 1 5 × 7 + …..
Find the sum of the series ∑ k = 1 360 1 k k + 1 + ( k + 1 ) k
Let a 1 , a 3 , … . a 10 be in A.P, and h 1 , h 2 , … h 10 be in H.P. If a 1 = h 1 = 2 a a 10 = h 10 = 3 , then a 4 h 7 is
If a, b, c are real and in A.P. and a 2 , b 2 , c 2 are in H.P., then
If a + b e y a − b e y = b + c e y b − c e y = c + d e y c − d e y they a , b , c , d are in
If the sum of m terms of an A.P is the same as the sum of its n terms, then the sum of its m + n term is
If a , b and c are in A.P, and b − a , c − b and a are in G.P, then a : b : c is
If S n denotes the sum of first n terms of an A.P whose first term is a and S n x S x is independent of x , then S ρ =
If the sum of n terms of an A.P. is cn ( n − 1 ) , where c ≠ 0 , then sum of the squares of these terms is
If a x 3 + b x 2 + c x + d is divisible by a x 2 + c , then a , b , c , d are in
The 10 th common term between the series 3 + 7 + 11 + … … … … . . and 1 + 6 + 11 + … … … . is
The minimum value of the expression 3 x + 3 1 − x , x ∈ R is:
If 1 , log 3 3 1 − x + 2 , log 3 4.3 x − 1 are in A.P then, x equals:
If 9 times the 9 th term of an A.P is equal to 13 times the 13 th term, then the 22 nd term of the A.P is:
Let S n denote the sum of the first n terms of an A.P. if S 2 n = 3 S n then S 3 n : S s =
Let T r be the r th term of an A.P. Whose first term is a and common difference is d . If for some positive integers m , n , m ≠ n , T m = 1 n and T n = 1 m and a − d equals:
Let α and β be the roots of the equation p x 2 + q x + r = 0 , p ≠ 0 . If p , q , r are in A.P., and 1 α + 1 β then the value of | α − β | is:
Let S 1 be the sum of the first n terms of the A.P. 8 , 12 , 16 … … … … and let S 2 be the sum of the first n terms of the A.P 17 , 19 , 21 … … … … , assume n ≠ 0 , then S 1 = S 2 for:
If 1 a + 1 a − 2 b + 1 c + 1 c − 2 b = 0 and a , b , c are not in A.P then:
If a 1 , a 2 , a 3 , … … . a n are in A.P with common difference d ≠ 0 , then sum of the series sin d sec a 1 sec a 2 + sec a 2 sec a 3 + … … … + sec a n − 1 sec a n is
The maximum sum of the series 20 + 19 1 3 + 18 2 3 + … … … . . is
a , b , c , d ∈ R such that a, b and c are in A.P and b , c and d are in H.P then
If x , y , z are positive integers then value of expression ( x + y ) ( y + z ) ( z + x ) is:
If x = ∑ n = 0 ∞ a n , y = ∑ n = 0 ∞ b n , z = ∑ n = 0 ∞ c n where a , b , c are in A.P and | a | < 1 , | b | < 1 , | c | < 1 , then x,y,z are in
If x ∈ R , the numbers 5 1 + x + 5 1 − x , a 2 , 25 x + 25 − x from an A.P then ‘a’ must lie in the imterval :
If 10 9 + 2 ( 11 ) 1 10 8 + 3 ( 11 ) 2 ( 10 ) 7 + . . + 10 ( 11 ) 9 = p ⋅ 10 9 then p is
The value of 1 + 1 1 2 + 1 2 2 + 1 + 1 2 2 + 1 3 2 + . . + 1 + 1 ( 2020 ) 2 + 1 ( 2021 ) 2 is
For a positive integer n, let a ( n ) = 1 + 1 2 + 1 3 + 1 4 + ⋯ + 1 2 n − 1 then
Let x = 2 2 2 2 … upto ∞ and y = 2 + 2 + 2 + 2 + … upto ∞ , then x-y equals
Sum of the series ∑ r = 1 n r ( r + 1 ) ! is
If 1 + 5 + 12 + 22 + 35 + … to n terms = n 2 ( n + 1 ) 2 n th term of series is
If θ 1 , θ 2 , θ 3 , … , θ n are in AP, whose common difference is d, then sin d sec θ 1 sec θ 2 + sec θ 2 sec θ 3 + … + sec θ n − 1 sec θ n is equal to
lf a, b, c are pth, qth and rth terms of a GP, then (q – r) log a + (r – p) log b + (p – q) log c is equal to
A man saved Rs 66000 in 20 yr. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?
The sum of the series 3 × 1 2 + 5 × 2 2 + 7 × 3 2 + … is
If a 1 , a 2 , a 3 , … , a 4001 are term of an AP such that 1 a 1 a 2 + 1 a 2 a 3 + … + 1 a 4000 a 4001 = 10 a 2 and a 2 + a 4000 = 50 , then a 1 − a 4001 is equal to
Between 1 and 31 are inserted M arithmetic means so that the ratio of the 7th and (M – 1) th means is 5:9, then the value of m is
if, a 2 , b 2 , c 2 are in AP, then which of the following is also in AP ?
Given sum of the first n terns of an AP is 2n + 3 n 2 . Another AP is formed with the same first term and double of the common difference, the sum of n terms of the new AP is
If log 2 ( 5 .2 x + 1 ) , log 4 ( 2 1 − x + 1 ) and 1 are in A.P, then x equals
If the sides of a right-angled triangle are in AP, then the sines of the acute angles are
Suppose a, b, c are in A.P. and a 2 , b 2 , c 2 , are in G.P . If a < b < c and a + b + c = 3 2 then the value of a, is
If the ratio of the sum of n terms of two AP’s be (7n + 1):(4n + 27), then the ratio of their 11th terms will be
If α and β are roots of the equation x 2 − 3 x + a = 0 and γ and δ are roots of the equation x 2 − 12 x + b = 0 and α , β , γ , δ form an increasing GP, then the values of a and b are respectively
The sum of the infinite series 1 + 4 3 + 9 3 2 + 16 3 3 + … ∞ is
1 + 3 2 + 5 2 2 + 7 2 3 … ∞ is equal to
Sum of n terms of series12+16+24+40+…will be
The sum of the series 1 + 3 x + 6 x 2 + 10 x 3 + … ∞ will be
If 1, log y x , log z y , − 15 log x z are in A.P., then
Four different integers form an increasing A.P. If one of these numbers is equal to the sum of the squares of the other three numbers, then the numbers are
If two geometric means g1 and g2 and one arithmetic mean A be inserted between two numbers, then g 1 2 g 2 + g 2 2 g 1 =
The number of numbers lying between 100 and 500 that are divisible by 7 but not by 21 is
If 5 1 + x + 5 1 − x , a 2 and 25 x + 25 − x are three consecutive terms of an A.P., then the values of a are given by
If log y x , log z y , − 15 log x z are in A.P., then
If three positive real numbers a, b, c are in A.P. such that abc = 4, then the minimum possible value of b is
If 1 4 13 + 2 4 35 + 3 4 5 .7 + … + n 4 ( 2 n − 1 ) ( 2 n + 1 ) = 1 48 f ( n ) + n 16 ( 2 n + 1 ) , then f (n) is equal to
The maximum sum of the series 20 + 19 1 3 + 18 2 3 + 18 + … is
The minimum number of terms from the beginning of the series 20 + 22 2 3 + 25 1 3 + … so that the sum may exceed 1568, is
A club consists of members whose ages are in A.P., the common difference being 3 months. If the youngest member of the club is just 7 years old and the sum of the ages of all the members is 250 years, then the number of members in the club are
The sum of all two digit numbers which when divided by 4, yield unity as remainder, is
If there are (2n + 1) terms in A.P., then the ratio of the sum of odd terms and the sum of even terms is
If a, b, c are in A.P. and p is the A.M. between a and b and q is the A.M. between b and c, then
Between two numbers whose sum is 2 1 6 , an even number of arithmetic means are inserted. If the sum of these means exceeds their number by unity, then the number of means are
If p, q, r are in A.P. and x, y, z are in G.P., then x q − r ⋅ y r − p ⋅ z p − q =
If, in a G.P., the (p + q)th term is a and the (p – q)th term is b, then pth term is
In a set of four numbers the first three are in G.P. and the last three are in A.P. with a common difference 6. If the first number is same as the fourth, the four numbers are
If the first term of an infinite G.P. is 1 and each term is twice the sum of the succeeding terms, then the common ratio is
Let a n be the nth term of the G.P. of positive numbers. Let ∑ n = 1 100 a 2 n = α and ∑ n = 1 100 a 2 n − 1 = β , such that α ≠ β , then the common ratio is
The sum S n to n terms of the series 1 2 + 3 4 + 7 8 + 15 16 + … is equal to
If p, q, r are in A.P. and x, y, z are in G.P., then x q − r ⋅ y r − p ⋅ z p − q =
The sum of series x 1 − x 2 + x 2 1 − x 4 + x 4 1 − x 8 + … to infinite terms, if f | x | < 1 is
If ( 1 − y ) 1 + 2 x + 4 x 2 + 8 x 3 + 16 x 4 + 32 x 5 = 1 − y 6 , ( y ≠ 1 ) , then a value of y x is
The largest value of the positive integer k for which n k + 1 divides 1 + n + n 2 + … + n 127 is divisible by
If the first term of an infinite G.P. is 1 and each term is twice the sum of the succeeding terms, then the common ratio is
If A = 1 + r a + r 2 a + r 3 a + … ∞ and If B = 1 + r b + r 2 b + r 3 b + … ∞ , then a b is equal to
If the sum of three numbers in G.P. is 63 and the product of the first and the second term is 3 4 of the third term, then the numbers are
If a, b, c are positive then the minimum value of a log b − log c + b log c − log a + c log a − log b is
If A denotes the property that two elements of A = {1, 5, 9, 13 …, 1093} add up to 1094, then the maximum number of elements in A can be
The sum of first n terms of the series 1 ⋅ 1 ! + 2 ⋅ 2 ! + 3 ⋅ 3 ! + 4 ⋅ 4 ! + … is
If three positive real numbers a, b, c are in A.P. such that abc = 4, then the minimum value of b is
Suppose a , b , c are positive and form an increasing G.P. If a + b + c = x b , then
If S n the sum of n terms of A.P, then S n + 3 − 3 S n + 2 + 3 S n + 1 − S n =
Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is
If a , 1 b , c and 1 p , q , 1 r form two arithmetic progressions of the same common difference, then a, q, c are in A.P. if
If S denotes the sum to infinity and S n , the sum of n terms of the series 1 + 1 2 + 1 4 + 1 8 + ⋯ , such that S − S n < 1 1000 then the least value of n is
Suppose a 1 , a 2 , … are in A.P. If a 8 : a 5 = 3 : 2, a 17 : a 23 is
Suppose A.M. of two positive numbers be 7 and G.M. between them be 5. The A.M. between their squares is
Sum of an infinite G.P. is 2 and sum of their cubes is 24, then 5th term of the G.P. is
If 7th term of an A.P. is 9 and 9th term of the A.P. is 7, then 20th term of the A.P. is
The sum to n terms of the series 1 7 + 10 + 1 10 + 13 + 1 13 + 16 + ⋯ is
If a 2 + b 2 , ab + bc and b 2 + c 2 are in G.P., then a, b, c are in
The interior angles of a convex polygon are in A.P. If the smallest angle is 100 ° and the common difference is 4 ° , then the number of sides is
Suppose m th term of an A.P. is 1/n and n th term of the A . P . is 1/m. If r th term of the A . P . is 1, then r is equal to
If the p th, q th and r th terms of an A.P. are in G.P., then the common ratio of the G.P. is
If S n = ∑ r = 1 n t r = 1 6 n 2 n 2 + 9 n + 13 , then ∑ r = 1 n t r equals
The number of terms of the A.P. 1, 4, 7, . . . that must be taken to obtain a sum of 715 is
If for every n ∈ N , , sum to n terms of an A.P. is 5 n 2 + 7 n then its 10th term is
If 1 1 2 + 1 2 2 + 1 3 2 + ⋯ upto ∞ = π 2 6 , then value of 1 1 2 + 1 3 2 + 1 5 2 + ⋯ up to ∞ is
If a and x are positive integers such that x < a and a – x , x , a + x are in A.P., then least possible value of a is
If X,Y, Z are real and 4 x 2 + 9 y 2 + 16 z 2 − 6 xy − 12 yz − 8 zx = 0 then x, y, z are in
If sum of first 20 terms of an A.P. is equal to sum of first 30 terms of the A.P. then sum of the first 50 terms of the A.P. is
Let f ( x ) be a polynomial function of second degree. If f ( 1 ) = f ( – 1 ) and a , b , c are in A.P., then f ‘ ( a ) , f ‘ ( b ) , f ‘ ( c ) are in
The digits of a three digit number N are in A.P. If sum of the digits is 15 and the number obtained by re-versing the digits of the number is 594 less than the original number, then 1000 N − 252 is equal to
If sum of four numbers in A.P. is 28 and product of two middle terms is 45, then product of the first and last terms is
If 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + ⋯ = π 4 , then value of 1 1 ⋅ 3 + 1 5 ⋅ 7 + 1 9 ⋅ 11 + ⋯ is
If the sum of three numbers in A.P. is 24 and their product is 440, then common difference of the A.P . can be
For x ∈ R , let x denote the greatest integer ≤ x . Largest natural number n for which E = π 2 + 1 100 + π 2 + 2 100 + π 2 … + n 100 + π 2 < 43 , is
n arithmetic means are inserted between 3 and 17. If the ratio of first to last mean is 1 : 3, then n is equal to:
Suppose a , b , c > 0 and p ∈ R . If a 2 + b 2 p 2 − 2 ( a b + b c ) p + b 2 + c 2 = 0 then a , b , c are in
Suppose ( m + n ) th term of a G . P . is p and ( m – n ) th term is q , then its n th is
If H 1 , H 2 , … , H n are n harmonic means between a and b ( ≠ a ) , then value of H 1 + a H 1 − a + H n + b H n − b is equal to
The number of terms of the G.P. 3 , 3 2 , 3 4 , . . . . . . needed to obtain a sum of 3069 512 is :
If the ratio of sums to n terms of two A.P.’s is ( 5 n + 7 ) : ( 3 n + 2 ) , then the ratio of their 17 t h terms is
Sum to 25 terms of the series 0.5 + 0.55 + 0.555 + . . . is:
If three positive real numbers a , b , c are in A.P. such that a b c = 4 , then the minimum possible value of b is
If one geometric G and two arithmetic means A 1 and A 2 are inserted between two distinct positive numbers, then 2 A 1 − A 2 G 2 A 2 − A 1 G equal to
For 0 < x < π the values of x which satisfy the relation 9 1 + | cos x | + cos 2 x + cos 3 x + ⋯ upto ∞ = 3 4 are given by
If sum of the infinite G.P. p + 1 + 1 p + 1 p 2 + … , ( p > 2 ) is 49/6, then sum of the 3rd term and the 4th term of the G.P. is
The sum to infinity of the series 1 + 4 5 + 7 5 2 + 10 5 3 + … is
The sum upto ( 2 n + 1 ) terms of the series a 2 − ( a + d ) 2 + ( a + 2 d ) 2 − ( a + 3 d ) 2 + … is
If the sum to n terms of an A.P. is 3 n 2 + 5 n ,while T m = 164 then value of m is
Value of y = ( 0.36 ) log 0.25 1 3 + 1 3 2 + 1 3 3 + … upto ∞ is
Sum to n terms of the series S = 1 + 2 1 + 1 n + 3 1 + 1 n 2 + ⋯ is given by
The value of x = 2 1 / 4 4 1 / 8 8 1 / 16 16 1 / 32 … is
If ( 20 ) 19 + 2 ( 21 ) ( 20 ) 18 + 3 ( 21 ) 2 ( 20 ) 17 + . . . + 20 ( 21 ) 19 = k ( 20 ) 19 then k is equal to
If a 1 , a 2 , a 3 , . . . . , a n , . . . are in A.P. such that a 4 – a 7 + a 10 = m , then sum of the first 13 terms of the A.P. is
If A 1 , A 2 be two arithmetic means and G 1 , G 2 be two geometric means between two positive numbers a and b, then A 1 + A 2 G 1 G 2 is equal to
Suppose a , b , c are positive real numbers. If a , b , c are in A.P. and a 2 , b 2 , c 2 are in H.P., then
If G1 and G2 are two geometric means and A is the arithmetic mean inserted between two positive numbers then the value of G 1 2 G 2 + G 2 2 G 1 is
The sum of first 26 terms of an A.P a 1 , a 2 , a 3 , … , If a 2 + a 6 + a 9 + a 18 + a 21 + a 25 = 165 , is
If 0 < θ , ϕ < π / 2 and x = ∑ n = 0 ∞ sin 2 n ϕ , y = ∑ n = 0 ∞ cos 2 n θ and z = ∑ n = 0 ∞ cos n ( θ + ϕ ) cos n ( θ − ϕ ) , then
The eighth term of a geometric progression is 128 and common ratio is 2. The product of the first five terms is
The sum of an infinite number of terms of a G.P. is 20, and the sum of their squares is 100, then the first term of the G.P. is
The first and last terms of an A.P. are a and l respectively. If s is the sum of all the terms of the A.P., then the common difference of the A.P. is
If l , m , n are positive and are respectively the p t h , q t h a n d r t h terms of a G.P., then Δ = log l p 1 log m q 1 log n r 1 is equal to
Three positive numbers form an increas-ing G.P. If the middle term in this G.P. is tripled, the new numbers are in A.P. Then the common ratio of G.P. is:
If a , b , c are in H.P. then a − b 2 , b 2 , c − b 2 are in
Let a 1 , a 2 , a 3 . . . . be terms of an A.P. If a 1 + a 2 + … + a p a 1 + a 2 + … + a q = p 2 q 2 , p ≠ q then a 6 a 21
Suppose m arithmetic means are inserted between 1 and 31. If the ratio of the second mean to the m th mean is 1 : 4, then m is equal to
Sum to n terms of the series 1 ( 1 + x ) ( 1 + 2 x ) + 1 ( 1 + 2 x ) ( 1 + 3 x ) + 1 ( 1 + 3 x ) ( 1 + 4 x ) + … is
If 1 2 + 2 2 + … + n 2 = 1015 , then value of n is
The sum to 10 terms of the series 2 + 6 + 18 + 54 + … is
If ( x ) < 1 , and r th term of a series is 1 + x + x 2 + … + x r − 1 , then sum to n terms of the series is
If a 1 , a 2 , … a n are in H.P. then the expression a 1 a 2 + a 2 a 3 + … + a n − 1 a n is equal to
1 2 + 1 2 + 2 2 + 1 2 + 2 2 + 3 2 + … upto 22nd terms is
In a geometric progression the ratio of the sum of the first 5 terms to the sum of their reciprocals is 49 and sum of the first and the third term is 35.The fifth term of the G.P. is
Let m be a positive integer, then S = ∑ k = 1 m k 1 k + 1 k + 1 + 1 k + 2 + … + 1 m is equal to :
If m th term of an A.P. is n and n th term in m, then its r th term is
Suppose a , b , c , are in A.P. and – 1 < a , b , c , Let x = ∑ n = 0 ∞ a n , y = ∑ n = 0 ∞ b n , z = ∑ n = 0 ∞ c n , then x , y , z are in
Suppose a, b, c are distinct real numbers. If a , b , c are in A.P. and a 2 , b 2 , c 2 are in H.P., then
In a G.P. consisting of positive terms, each term equals the sum of the next two terms. Then common ratio of the G.P. is
If the sum of the series 2 + 5 x + 25 x 2 + 125 x 3 + … is finite, then
Sum of the series S = 1 2 − 2 2 + 3 2 − 4 2 + . . . – 2008 2 + 2009 2 is
Let f ( n ) = 1 5 + 3 n 100 n , where ( x ) denotes the greatest integer less than or equal to x . Then ∑ n = 1 61 f ( n )
If a, b, c are positive real numbers that are in G.P., then the equations a x 2 + 2 b x + c = 0 and d x 2 + 2 e x + f = 0 have a common root if a / d , b / e , c / f are in
If log 5 2 , log 5 2 x − 5 and log 5 2 x − 7 / 2 are in A.P., then x is equal to
If log 10 2 , log 10 2 x + 1 and log 10 2 x + 3 are in A.P., then
The value of n for which 704 + 1 2 ( 704 ) + 1 4 ( 704 ) + … up n terms = 1984 − 1 2 ( 1984 ) + 1 4 ( 1984 ) – . . . . up to n terms is
The positive integer n for which 2 × 2 2 + 3 × 2 3 + 4 × 2 4 + … + n × 2 n = 2 n + 10 is
If log ( a + c ) + log ( a + c − 2 b ) = 2 log ( a − c ) then
Sum of the series S = 1 + 1 2 ( 1 + 2 ) + 1 3 ( 1 + 2 + 3 ) + 1 4 ( 1 + 2 + 3 + 4 ) + … up to 20 terms is
The n t h term of the sequence 2 1 2 , 1 7 13 , 1 1 9 , 20 23 . is
Sum to n terms of the series tan − 1 1 3 + tan − 1 1 7 + tan − 1 1 13 + … is
If H n = 1 + 1 2 + 1 3 + … + 1 n then value of 1 + 3 2 + 5 3 + … + 2 n − 1 n is
H.M. between 1 28 and 1 10 is
Sum to 30 terms of the series 1 1.2.3 + 1 2.3.4 + 1 3.4.5 + … is
If a 1 , a 2 , … , a n are in A.P., with common difference d ≠ 0 then sum of the series sin d cosec a 1 cosec a 2 + cosec a 2 cosec a 3 + … + cosec a n − 1 cosec a n is
Coefficient of x 99 in the expansion of ( x − 1 ) ( x − 3 ) ( x − 5 ) … ( x − 199 ) is
Let T r be the rth term of an AP, for r = 1 , 2…. . If for some positive integers m and n, we have T m = 1 n and T n = 1 m , the T m + n equal
If L = lim n ∞ 1 + 3 − 1 1 + 3 − 2 1 + 3 − 4 1 + 3 − 8 … 1 + 3 − 2 n , then
Sum to n terms of the series 1 + ( 1 + 2 ) + ( 1 + 2 + 3) + … is
Fifth term of a G.P. is 2, then the product of its 9 terms is
For a positive integer n , let a ( n ) = 1 + 1 2 + 1 3 + 1 4 + ⋯ + 1 2 n − 1 . Then
If s is the sum to infinity of a G.P., whose first term is a, then the sum of the first n terms of the G.P. is
If x > 1 , y > 1 , z > 1 are in G.P. , then 1 1 + ln x , 1 1 + ln y , 1 1 + ln z are in
Let a n be the n t h term of an A.P. If ∑ r = 1 100 a 2 r = α and ∑ r = 1 100 a 2 r − 1 = β , then the common difference of the A.P. is:
Let x be the arithmetic mean and y , z be the two geometric means between any two positive numbers. Then value of y 3 + z 3 x y z is
The sum to n terms of the series 1 1 3 + 1 + 2 1 3 + 2 3 + 1 + 2 + 3 1 3 + 2 3 + 3 3 + … is
The H.M. between two numbers is 16/5, their A.M. is A and G.M. is G. If 2 A + G 2 = 26 then the numbers are
Statement – 1 : The sum of the series 1 + ( 1 + 2 + 4 ) + ( 4 + 6 + 9 ) + ( 9 + 12 + 16 ) + … + ( 361 + 380 + 400 ) is 8000 Statement – 2 : ∑ k = 1 n k 3 − ( k − 1 ) 3 = n 3 for each natural number n .
Consider an A.P. a 1 , a 2 , a 3 , … such that a 3 + a 5 + a 8 = 11 and a 4 + a 2 = − 2 then the value of a 1 + a 6 + a 7 is
If 22 7 and π appear as two distinct terms of an A.P., then common difference of the A.P. must be
The value of 0 · 2 log 5 1 4 + 1 8 + 1 16 + ⋯ is
Let I n = ∫ 0 π / 4 tan n x d x . Then I 2 + I 4 , I 3 + I 5 , I 4 + I 6 , I 5 + I 7 , … are in
If a , x , b are in H.P. and a , y , z , b are in G.P., then the value of y z x y 3 + z 3 is
The sum of three numbers in G.P. is 14. If one is added to the first and second numbers and I is subtracted from the third, the new numbers are in A.P. The smallest of them is
a , b , c , d ∈ R + such that a, b and c are in A.P. and b, c and d are in H.P., then
The value of the sum ∑ i = 1 20 i 1 i + 1 i + 1 + 1 i + 2 + … + 1 20 is
Let a , b and c be distinct real numbers. If a , b , c are in geometric progression and a + b + c = x b , then x lies in the set
The sum of the numbers between 200 and 400 that are divisible by 7 is
If log 10 2 , log 10 2 x − 1 and log 10 2 x + 3 are three consecutive terms of an A.P. for
Let a , b , c , d and e be distinct positive number. If a , b , c and 1 c , 1 d , 1 e both are in A.P. and b , c , d are in G.P. then
If ∑ k = 1 n ϕ ( k ) = 2 n n + 1 , then ∑ k = 1 10 1 ϕ ( k ) is equal to
The sum of n terms of an A . P . is 3 n 2 + 5 The number of term which equals 159, is
If a 1 , a 2 , a 3 is an A . P . such that a 1 + a 5 + a 10 + a 15 + a 20 + a 24 = 225 then a 1 + a 2 + a 3 + … … + a 23 + a 24 is equal to
