First slide

Special Series in Sequences and Series

Question

The series of natural numbers is divided into groups $1; 2, 3, 4; 5, 6, 7, 8, 9; ......$ and so on. Then the sum of the numbers in the nth group is

Difficult

A

$(2 n - 1) (n^{2} - n + 1)$
B

$n^{3} - 3 n^{2} + 3 n - 1$
C

$n^{3} + {(n - 1)}^{3}$
D

$n^{3} + {(n + 1)}^{3}$

Solution

Last term ${(n - 1)}^{t h} g r o u p i s {(n - 1)}^{2}$
$∴$ first term is n^th group is ${(n - 1)}^{2} + 1$ and last term of n^th group is $n^{2}$
Number of terms in n^th group $= (2 n - 1)$
$∴$ Sum of the numbers in the n^th group $= \frac{(2 n - 1)}{2} {{(n - 1)}^{2} + 1 + n^{2}}$
                                                                          $= (2 n - 1) (n^{2} - n + 1) = 2 n^{3} - 2 n^{2} + 2 n - n^{2} + n - 1$
   $= 2 n^{3} - 3 n^{2} + 3 n - 1$
                                                                           $= n^{3} + {(n - 1)}^{2}$

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