Question

The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28^th term of this A.P.

Answer 1

Let a be the first term and d be the common difference.

We know that, sum of first n terms = S_n = \[\frac{n}{2}\][2a + (n − 1)d] 

It is given that sum of the first 7 terms of an A.P. is 63.

And sum of next 7 terms is 161.

∴ Sum of first 14 terms = Sum of first 7 terms + Sum of next 7 terms

= 63 + 161 = 224

Now,

S₇ = \[\frac{7}{2}\][2a + (7 − 1)d]

⇒ 63 = \[\frac{7}{2}\] (2a + 6d)

⇒ 18 = 2a + 6d

⇒ 2a + 6d = 18         ....(1) 

Also,

S₁₄ =  \[\frac{14}{2}\][2a + (14 − 1)d]

⇒ 224 = 7(2a + 13d)

⇒ 32 = 2a + 13d

⇒ 2a + 13d =  32      ....(2)

On subtracting (1) from (2), we get

13d − 6d = 32 − 18

⇒ 7d = 14

⇒ d = 2

⇒ 2a = 18 − 6d         [From (1)]

⇒ 2a = 18 − 6 × 2

⇒ 2a = 18 − 12

⇒ 2a = 6

⇒ a = 3

Also, n^th term = a_n = a + (n − 1)d

⇒ a₂₈ = 3 + (28 − 1)2

= 3 + 27 × 2

= 57

Thus, 28^th term of this A.P. is 57.