Question

The sum of $5^{th}$ and $9^{th}$ terms of an A.P. is $30$. If its ${25}^{th}$ terms is three times its $8^{th}$ term, then find the A.P.

Answer 1

Let $a=$ first term of A.P., $d=$ common difference $n^{th}$ term, $a_n=a+(n-1)d$;

Given, $a_5+a_9-30$

$\Rightarrow a+4d+a+8d=30$

$\Rightarrow 2a+12d=30$

$\Rightarrow a+6d=15$;

$a_{25}=3a_8$

$\Rightarrow a+24d=3(a+7d)$

$\Rightarrow a+24d=3a+21d$

$\Rightarrow 2a=3d$

$\Rightarrow a=\frac{3d}{2}$

Putting in $\left(1\right)$

$\frac{3d}{2}+6d=15$

$\Rightarrow 15d=15\times 2$

$d=2$

$\Rightarrow a=3$.

A.P. : $3,5,7,9,$…..