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, S_{5} = 25

⇒ 2*a* + 4*d* = 10 ........ (1)

Given, S_{1}+ S_{2} + ....... + S_{5} = 25 and

S_{6 }+ S_{7} + ....... + S_{10} = –75

⇒ S_{1 }+ S_{2} + ....... + S_{9} + S_{10 }= 25 –75 = –50

⇒ S_{10} = – 50

5 [ 2*a* + 9*d*] = – 50

2*a* + 9*d *= – 10 .........(2)

Subtract (1) from (2), we get

2*a* + 9*d *–2*a* – 4*d = *– 10 – 10

5*d* = – 20

⇒*d* = – 4

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

2*a* + 4(–4) = 10

2*a* – 16 = 10

2*a* = 26

⇒ *a* = 13

∴ 10^{th} term of A.P. = a_{10} = *a* + 9*d*

= 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

a_{16} = *a* + (*n* – 1) *d*

= 8 + (16 – 1) × 7

= 8 + 15 × 7 = 113

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