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Q)

Find sum upto n terms: $\large\frac{3}{1^2.2^2}+\large\frac{5}{2^2.3^2}+\large\frac{7}{3^2.4^2}$+....

$(a)\;\large\frac{n(n+2)}{(n+1)^2}\qquad(b)\;\large\frac{n^2(n+2)}{(n+1)}\qquad(c)\;\large\frac{n(n+2)^2}{n+1}\qquad(d)\;None$

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A)
Answer : (a) $\;\large\frac{n(n+2)}{(n+1)^2}$
Explanation : $a_{r}=\large\frac{2r+1}{r^2.(r+1)^2}$
$=\large\frac{(r+1)^2-r^2}{r^2\;.(r+1)^2}$
$=\large\frac{1}{r^2}-\large\frac{1}{(r+1)^2}$
$S_{n}=\sum_{r=1}^{n}\;a_{r}$
$=\sum_{r=1}^{n}\;\large\frac{1}{r^2}-\large\frac{1}{(r+1)^2}$
$S_{n}=1-\large\frac{1}{(n+1)^2}$
$=\large\frac{(n+1)^2-1}{(n+1)^2}$
$S_{n}=\large\frac{n(n+2)}{(n+1)^2}\;.$
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