Question

The sum of the series $\frac{5}{1^{2} \cdot 4^{2}}+\frac{11}{4^{2} \cdot 7^{2}}+\frac{17}{7^{2} \cdot 10^{2}}+.\ldots \ldots .$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $2$
• D. $3$

Answer 1

Answer: (B)

$S=\frac{5}{1^{2} \cdot 4^{2}}+\frac{11}{4^{2} \cdot 7^{2}}+\frac{17}{7^{2} \cdot 10^{2}}+.\ldots $ $3S=\frac{3.5}{1^{2} \cdot 4^{2}}+\frac{3 \cdot 11}{4^{2} \cdot 7^{2}}+\frac{3 \cdot 17}{7^{2} \cdot 10^{2}}+.\ldots .$
$3S=\frac{\left(4 - 1\right) \left(4 + 1\right)}{1^{2} \cdot 4^{2}}+\frac{\left(7 - 4\right) \left(7 + 4\right)}{4^{2} \cdot 7^{2}}+\frac{\left(10 - 7\right) \left(10 + 7\right)}{7^{2} \cdot \left(10\right)^{2}}+.\ldots .$
$3S=\frac{4^{2} - 1^{2}}{1^{2} \cdot 4^{2}}+\frac{7^{2} - 4^{2}}{4^{2} \cdot 7^{2}}+\frac{10^{2} - 7^{2}}{7^{2} \cdot 10^{2}}+.\ldots .$
$=\frac{1}{1^{2}}-\frac{1}{4^{2}}+\frac{1}{4^{2}}-\frac{1}{7^{2}}+\frac{1}{7^{2}}-\frac{1}{10^{2}}+.\ldots .$
$\Rightarrow 3S=1$
$S=\frac{1}{3}$