Question

Find the common difference of an $AP$ whose First term is $5$ and the sum of the First Four terms is half the sum of next four terms.

Answer 1

Let $d$ be the common difference of given A.P.

First term $\left(a\right)=5$

Given:-

$a_1+a_2+a_3+a_4=\frac{1}{2}(a_5+a_6+a_7+a_8)$

$\Rightarrow a+(a+d)+(a+2d)+(a+3d)=\frac{1}{2}[(a+4d)+(a+5d)+(a+6d)+(a+7d)]$

$\Rightarrow 4a+6d=\frac{1}{2}(4a+22d)$

$\Rightarrow 4a+6d=2a+11d$

$\Rightarrow 11d-6d=4a-2a$

$\Rightarrow d=\frac{2\times 5}{5}=2$

Hence the common difference of given A.P. is $2$.