Question

The value of $\frac{5}{1^2{4}^2}+\frac{11}{4^2{7}^2}+\frac{5}{7^2{10}^2}+......\infty$ is equal to

• A. $1$
• B. $\frac{1}{3}$
• C. $\frac{1}{5}$
• D. $\frac{1}{7}$

Answer 1

The correct option is B $\frac{1}{3}$

Let $S=\frac{5}{1^2{4}^2}+\frac{11}{4^2{7}^2}+\frac{17}{7^2{10}^2}+...$

Multiplying with $3$ on both sides

$3S=\frac{15}{1^2{4}^2}+\frac{33}{4^2{7}^2}+\frac{51}{7^2{10}^2}+...3S=\frac{1}{1^2}-\frac{1}{4^2}+\frac{1}{4^2}-\frac{1}{7^2}+\frac{1}{7^2}-\frac{1}{{10}^2}+...\infty 3S=1S=\frac{1}{3}$