Question

In an arithmetic progression of which $a$ is the first term. If the sum of the first ten terms is zero and the sum of next ten terms is $-200$ then find the value of $a$.

Answer 1

First term=a

Let the common difference $=$d

Sum of first ten terms$=$0

Sum of first n term of an AP $=$ $\frac{n}{2}(2a+(n-1\left)d\right)$

By given $S_{10}=0$

$\frac{10}{2}(2a+(10-1\left)d\right)=0$

$\frac{10}{2}(2a+9d)=0$

$2a+9d=0$----- (Equation 1)

Sum of next ten terms$=-200$

Sum of first ten terms+Sum of next ten terms$=0-200$

Sum of first ten terms+Sum of next ten terms$=-200$

So,

Sum of first twenty terms $S_{20}$$=-200$

$\frac{20}{2}(2a+(20-1\left)d\right)=-200$

$\frac{20}{2}(2a+19d)=-200$

$2a+19d=-20$ ------ (Equation 2)

From equation 1,

$2a=-9d$

Put the value of $2a$ in equation 2,

$-9d+19d=-20$

$10d=-20$

$d=-2$

Put the value of "$d$" in equation 1,

$2a-18=0$

$2a=18$

$a=9$