Question

The first and the last terms of an A.P. are $a$ and $l$ respectively. If $S$ be the sum of all the terms of the A.P., then the common difference is

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

Answer 1

Answer: (C)

$\frac{l^2-a^2}{2S-l-a}$

Let the terms of A.P. are
$a,a+d,a+2d,...a+(n-1)d$
Given $S=\frac{n}{2}(a+l)\therefore n=\frac{2S}{a+l}$
Again $l=a+(n-1)d$
$\therefore l=a+(\frac{2S}{a+l}-1)d$
$\therefore d=\frac{(l-a)(l+a)}{2S-1-a}$
$d=\frac{l^2-a^2}{2S-l-a}$