Banner 0
Banner 1
Banner 2
Banner 3
Banner 4
Banner 5

Q.

An A.P. consists of 37 terms. The sum of the three middle most terms is 225

and the sum of the last three terms is 429. Find the A.P.

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!
An Intiative by Sri Chaitanya

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

We are given the sum of the three middlemost terms of an A.P. and the sum of the last three terms, and we have to find the A.P.
The A.P. has 37 terms, so ๐‘› = 37.
Let the first term of the A.P. be ๐‘Ž and the common difference be ๐‘‘. The middlemost term of the A.P. is = ๐‘›+1/2 
37+1/ 2
38/ 2
= 19
So, the 19๐‘กโ„Ž term is the middlemost term.
The sum of the three middlemost terms = 225
The three middlemost terms are the 18๐‘กโ„Ž, 19๐‘กโ„Ž and 20๐‘กโ„Ž terms. 

So, ๐‘Ž18 + ๐‘Ž19 + ๐‘Ž20 = 225
โ‡’ ๐‘Ž + (18 โˆ’ 1)๐‘‘ + ๐‘Ž + (19 โˆ’ 1)๐‘‘ + ๐‘Ž + (20 โˆ’ 1)๐‘‘ = 225
โ‡’ 3๐‘Ž + 17๐‘‘ + 18๐‘‘ + 19๐‘‘ = 225
โ‡’ 3๐‘Ž + 54๐‘‘ = 225
โ‡’ 3(๐‘Ž + 18๐‘‘) = 225
โ‡’ ๐‘Ž + 18๐‘‘ = 225/3
โ‡’ ๐‘Ž + 18๐‘‘ = 75                               ....(1)
The sum of the last three terms = 429
The three last terms are the 35๐‘กโ„Ž, 36๐‘กโ„Ž and 37๐‘กโ„Ž terms. 

So, ๐‘Ž35 + ๐‘Ž36 + ๐‘Ž37 = 429
โ‡’ ๐‘Ž + (35 โˆ’ 1)๐‘‘ + ๐‘Ž + (36 โˆ’ 1)๐‘‘ + ๐‘Ž + (37 โˆ’ 1)๐‘‘ = 429
โ‡’ 3๐‘Ž + 34๐‘‘ + 35๐‘‘ + 36๐‘‘ = 429
โ‡’ 3๐‘Ž + 105๐‘‘ = 429
โ‡’ 3(๐‘Ž + 35๐‘‘) = 429
โ‡’ ๐‘Ž + 35๐‘‘ = 429/3 
โ‡’ ๐‘Ž + 35๐‘‘ = 143                               ....(2)
Subtracting eq(1) from eq(2), we get
๐‘Ž + 35๐‘‘ โˆ’ (๐‘Ž + 18๐‘‘) = 143 โˆ’ 75
โ‡’ ๐‘Ž + 35๐‘‘ โˆ’ ๐‘Ž โˆ’ 18๐‘‘ = 68
โ‡’ 17๐‘‘ = 68
โ‡’ ๐‘‘ = 68/17
โ‡’ ๐‘‘ = 4
So, the common difference between the A.P. is 4. Putting the value of ๐‘‘ in eq(1), we get
๐‘Ž + 18 ร— 4 = 75
โ‡’ ๐‘Ž + 72 = 75
โ‡’ ๐‘Ž = 75 โˆ’ 72

โ‡’ ๐‘Ž = 3

So, the first term of the A.P. is 3.

Hence, the A.P. can be written as ๐‘Ž, ๐‘Ž + ๐‘‘, ๐‘Ž + 2๐‘‘, ๐‘Ž + 3๐‘‘,..., ๐‘Ž + 36๐‘‘

= 3, 7, 11, 15, ..., 147.

Watch 3-min video & get full concept clarity

Best Courses for You

JEE

JEE

NEET

NEET

Foundation JEE

Foundation JEE

Foundation NEET

Foundation NEET

CBSE

CBSE

score_test_img

Get Expert Academic Guidance โ€“ Connect with a Counselor Today!

whats app icon
An A.P. consists of 37 terms. The sum of the three middle most terms is 225and the sum of the last three terms is 429. Find the A.P.