Q) The sum of the first five terms of an AP is 25 and the sum of its next five terms is -75.Find the 10th term of the AP?

Q)In an AP the first term is 8 and the common difference is 7.If the last term of the AP is 218,find its middle term?

1.

Let a and d the first term and common difference of given A.P.

According to given condition, S₅ = 25

⇒ 2a + 4d = 10  ........ (1)

Given, S₁+ S₂ +  .......  + S₅ = 25 and

S₆ + S₇ +  .......  + S₁₀ = –75

⇒ S₁ + S₂ +  .......  + S₉ + S₁₀  = 25 –75 = –50

⇒ S₁₀ = – 50

5 [ 2a + 9d] = – 50

2a + 9d = – 10  .........(2)

Subtract (1) from (2), we get

2a + 9d –2a – 4d = – 10 – 10

5d = – 20

⇒d = – 4

On putting value of d in (1), we get

2a + 4(–4) = 10

2a – 16 = 10

2a = 26

⇒ a = 13

∴ 10^th term of A.P. = a₁₀ = a + 9d

= 13 + 9 (– 4)

= – 23

(2)

Given, first term = a = 8

Common difference = d = 7 and

last term of A.P. = 218

Let there be n terms in given A.P.

∴ a_n = 218

⇒ a + (n – 1) d = 218

8 + (n – 1) 7 = 218

(n – 1) 7 = 210

n – 1 = 30

⇒ n = 31

The middle term is given by

a₁₆ = a + (n – 1) d

= 8 + (16 – 1) × 7

= 8 + 15 × 7 = 113