Question

find the common difference of an A.P. whose first term is 5 and the sum of first four terms is half the sum of next four terms.

Answer 1

let d is common difference of AP
now first 4term is 5,5+d,5+2d,5+3d
and next 4term 5+4d,5+5d,5+6d,5+7d
now according to question
20+6d=(20+22d)/2
20+6d=10+11d
d=2

Answer 2

2 is the common difference of an AP

Given:

a (first term of the arithmetic progression) = 5

$S_{4}=\frac{1}{2}\left(S_{8}-S_{4}\right)$

To find:

d (Common Difference) = ?

Solution:

The general sequence of an AP is a ,a + d ,a + 2d ,a + 3d,…

Substituting a=5 then  

5, 5 + d,5 + 2d,5 + 3d,5 + 4d,5 + 5d,5 + 6d,5 + 7d,,..

Let the first 4 terms be 5,5 + d,5 + 2d,5 + 3d

And let the next 4 terms be = 5 + 4d,5 + 5d,5 + 6d,5 + 7d  

And $\bold{S_{4}=\frac{1}{2}\left(S_{8}-S_{4}\right)----(1)}$

By substituting these values in (1)

$\begin{array}{l}{5+5+d+5+2 d+5+3 d} \\ {\qquad \qquad=\frac{1}{2}(5+4 d+5+5 d+5+6 d+5+7 d)}\end{array}$

20+6d=10+11d

10=5d

d=2

Therefore, the common difference = 2