Question

In an A.P. the sum of first $11$ terms is $44$ and that of the next $11$ terms is $55$. Find the first term and the common difference.

Answer 1

Given, $S_{11}=44$ and $S_{22}=44+55=99$

Formula for A.P. is

(i) $S_n=\frac{1}{2}\times n(a+l)$,

where $n$ is $n^{th}$ term, $a$ is $1^{st}$ term, $d$ is the common differnce and

$l$ is the last term which is given by $l=a+(n-1)d$

Using formula (i) for $n=11$ we get,

$S_{11}=\frac{1}{2}\times 11(a+l)d)$

$⟹$ $44=\frac{1}{2}\times 11(2a+(11-1\left)d\right)$ ..... by using the value of $l$

$⟹$ $\frac{44\times 2}{11}=2a+10d$

$⟹$ $2a+10d=8$ ............ $\left(1\right)$

Using the same formula for $n=22$,

$S_{22}=\frac{1}{2}\times 22(2a+(22-1\left)d\right)$

$⟹$ $99=\frac{1}{2}\times 22(2a+21d)$

$⟹$ $\frac{99\times 2}{22}=2a+21d$

$⟹$ $2a+21d=9$ ............ $\left(2\right)$

Solving equations $\left(1\right)$ and $\left(2\right)$ simultaneously we get,

$11d=1$ $⟹$ $d=\frac{1}{11}$

Substituting value of $d$ in equation $\left(1\right)$ we get,

$a=\frac{39}{11}$

Hence, first term $a=\frac{39}{11}$ and common difference $d=\frac{1}{11}$