# 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

∴ 10th 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.

an = 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

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