Question

In an arithmetic series, the sum of first $14$ terms is $-203$ and the sum of the next $11$ terms is $-572$. Find the arithmetic series

Answer 1

Given that $S_{14}=-203$
$\Rightarrow \frac{14}{2}[2a+13d]=-203$
$\Rightarrow 7[2a+13d]=-203$
$\Rightarrow 2a+13d=-29....\left(1\right)$.
Also, the sum of the next $11$ terms $=-572$.
Now, $S_{25}=S_{14}+(-572)$
That is, $S_{25}=-203-572=-775$
$\Rightarrow \frac{25}{2}[2a+24d]=-775$
$\Rightarrow 2a+24d=-31\times 2$
$\Rightarrow a+12d=-31....\left(2\right)$
Solving $\left(1\right)$ and $\left(2\right)$ we get, $a=5$ and $d=-3$.
Thus, the required arithmetic series is $5+(5-3)+(5+2(-3\left)\right)+.....$
That is, the series is $5+2-1-4-7-.....$