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  4. The value of S=(5/12 ⋅ 42)+(11/42 ⋅ 72)+(17/72 ⋅ 102)+⋅s ∞ is

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Q. The value of $S=\frac{5}{1^{2} \cdot 4^{2}}+\frac{11}{4^{2} \cdot 7^{2}}+\frac{17}{7^{2} \cdot 10^{2}}+\cdots \infty$ is

Sequences and Series

Solution:

$S=\frac{5}{1^{2} \cdot 4^{2}}+\frac{11}{4^{2} \cdot 7^{2}}+\frac{17}{7^{2} \cdot 10^{2}}+\cdots \infty$
$3 S=\frac{3 \cdot 5}{1^{2} \cdot 4^{2}}+\frac{3 \cdot 11}{4^{2} \cdot 7^{2}}+\frac{3 \cdot 17}{7^{2} \cdot 10^{2}}+\cdots \infty$
$\Rightarrow 3 S=\frac{(4-1)(4+1)}{1^{2} \cdot 4^{2}}+\frac{(7-4)(7+4)}{4^{2} \cdot 7^{2}}$
$+\frac{(10-7)(10+7)}{7^{2} \cdot 10^{2}}+\cdots \infty$
$\Rightarrow S=\frac{1}{3}$