Question

The first and last terms of an $AP$ are $a$ and $\ell$ respectively. If $s$ be the sum of all the terms of the $AP$, then common difference is

• A. $\frac{{\ell }^2-a^2}{2S-(\ell +a)}$
• B. $\frac{{\ell }^2-a^2}{2S-(\ell -a)}$
• C. $\frac{{\ell }^2+a^2}{2S+(\ell +a)}$
• D. $\frac{{\ell }^2+a^2}{2S-(\ell +a)}$

Answer 1

Answer: (A)

$\frac{{\ell }^2-a^2}{2S-(\ell +a)}$

Let $n$ be the number of terms and $s_n$be the sum of n terms in given AP.
Since, $l=a+(n-1)d$
$\Rightarrow l-a=(n-1)d$
$\Rightarrow \frac{l-a}{n-1}=d$ .....(i)
we know $s_n=\frac{n}{2}[a+l]$
$\therefore s=\frac{n}{2}[a+l]$
$\Rightarrow 2s=n(a+l)$
$\Rightarrow n=\frac{2s}{a+l}$
Now put $n=\frac{2s}{a+l}$ in equation (i)
$\Rightarrow \frac{l-a}{\frac{2s}{a+l}-1}=d$
$\Rightarrow \frac{l-a}{\frac{2s-(a+l)}{a+l}}=d$
$\Rightarrow \frac{(l-a)(l+a)}{2s-(a+l)}=d$
$\Rightarrow d=\frac{l^2-a^2}{2s-(l+a)}$
Option A is correct.