Question

If $S_r$ denotes the sum of $r$ terms of an AP and $\frac{S_a}{a^2}=\frac{S_b}{b^2}=c$ then $S_c$ is

• A. $c^3$
• B. $c/ab$
• C. $abc$
• D. $a+b+c$

Answer 1

The correct option is A $c^3$

$S_r$ denotes sum of $r$ terms.

$\frac{S_a}{a^2}=\frac{S_b}{b^2}\frac{\frac{a}{2}(2a_1+(a-1\left)d\right)}{a^2}=\frac{\frac{b}{2}(2a_1+(b-1\left)d\right)}{b^2}\frac{2a_1}{a}+\frac{(a-1)d}{a}=\frac{2a_1}{b}+\frac{(b-1)d}{b}or,\frac{2a_1}{a}+d-\frac{d}{a}=\frac{2a_1}{b}+d-\frac{d}{b}or,2a_1(\frac{1}{a}-\frac{1}{b})=d(\frac{1}{a}-\frac{1}{b})So,\frac{a}{2}\left(\frac{2a_1+(a-1)d}{a^2}\right)=cor,2a_1\frac{(1+a-a)}{2a}=c\frac{a_{1}a}{a}=cc=a_1{S}_c=\frac{c}{2}(2a_1+(c-1\left)d\right)=\frac{c}{2}(2a_1+(c-1\left)2a_1\right)=\frac{c}{2}(2c+(c-1\left)2c\right)=\frac{c}{2}\times 2c(1+c-1)=c^3$

Answer $\left(A\right)$