Question

The sum of 8 terms of an A.P. is 12, and the sum of 16 terms is 56.
Find the AP

Answer 1

Explanation:

`S_(n)=(n)/(2)(2a + (n-1)*d)`

Here, n- number of terms ; d - common difference ; a - first term

`S_(8)=12\\\\(8)/(2)[2a+(8-1)*d]=12\\\\4[2a+7d]=12\\\\2a + 7d = (12)/(4)\\\\\\2a + 7d = 3 \ -----------------------(i)`

`S_(16)=56\\\\(16)/(2)(2a+15d)=56\\\\\\8(2a+15d)=56\\\\2a + 15d=(56)/(8)\\\\\\`

2a + 15d = 7 ----------------(ii)

Subtract (i) from equation (ii)

(ii) 2a + 15d = 7

(i) 2a + 7d = 3

- - -

8d = 4

d = 4/8

d = 1/2

Plugin d = 1/2 in equation (i)

`2a +7*(1)/(2)=3\\\\\\2a = 3 -(7)/(2)\\\\\\2a =(6)/(2)-(7)/(2)\\\\\\2a=(-1)/(2)\\\\\\a=(-1)/(2*2)\\\\\\a=(-1)/(4)`

Second term = first term + d

`=(-1)/(4)+(1)/(2)=(-1)/(4)+(2)/(4)=(1)/(4)`

`Third \ term =(1)/(4)+(1)/(2)=(1)/(4)+(2)/(4)=(3)/(4)\\\\\\Fourth \ term =(3)/(4)+(1)/(2)=(3)/(4)+(2)/(4)=(5)/(4)\\\\`

AP is :

`(-1)/(4); (1)/(4);(3)/(4);(5)/(4).......`