Question

If $a$ and $l$ be the first and last term of an A.P. and $S$ be the sum of its all terms, then its 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. None of these

Answer 1

The correct option is B $\frac{l^2-a^2}{2S-l-a}$

Let $S$ be the sum of its all terms

$S=\frac{n}{2}(a+l)$

$\Rightarrow 2S=n(a+l)$

$\Rightarrow n=\frac{2S}{a+l}$

If $a$ and $l$ be the first and last term of an A.P.then

$l=a+(n-1)d$

$\Rightarrow l-a=(n-1)d$

$\Rightarrow d=\frac{l-a}{n-1}$

$\Rightarrow d=\frac{l-a}{\frac{2S}{a+l}-1}$ where $n=\frac{2S}{a+l}$

$\Rightarrow d=\frac{(l-a)(a+l)}{2S-a-l}$

$\Rightarrow d=\frac{(l-a)(l+a)}{2S-a-l}$

$\Rightarrow d=\frac{(l^2-a^2)}{2S-a-l}$