Question

Calculate: $\frac{a-b}{a+b}+\frac{3a-2b}{a+b}+\frac{5a-3b}{a+b}....$ to $11$ terms

• A. $\frac{111a-b}{a+b}$
• B. $\frac{121a-12b}{a+b}$
• C. $\frac{11(11a-6b)}{a+b}$
• D. $\frac{18(a-b)}{a+b}$

Answer 1

The correct option is C $\frac{11(11a-6b)}{a+b}$

Here first term A=$\frac{a-b}{a+b}$ and common difference

D=$\frac{3a-2b}{a+b}-\frac{a-b}{a+b}=\frac{2a-b}{a+b}$

$S_{11}=\frac{11}{2}(2A+(11-1\left)D\right)=11(A+5D)$

$\Rightarrow S_{11}=11(\frac{a-b}{a+b}+5\times \frac{2a-b}{a+b})=11(\frac{a-b}{a+b}+\frac{10a-5b}{a+b})=11(\frac{11a-6b}{a+b})$

$\Rightarrow S_{11}=\frac{121a-66b}{a+b}$