1 3^ 2 -1 +1 5^ 2 -1 +1 7^ 2 -1 + +1 (201)^ 2 -1 is equal to - Vidyadip

MCQ

$\frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+\ldots+\frac{1}{(201)^{2}-1}$ is equal to

A
$\frac{101}{404}$
✓
$\frac{25}{101}$
C
$\frac{101}{408}$
D
$\frac{99}{400}$

✓

Answer

Correct option: B.

$\frac{25}{101}$

b
$T_{n}=\frac{1}{(2 n+1)^{2}-1} \frac{1}{(2 n+2) 2 n}=\frac{1}{4(n)(n+1)}$

$=\frac{(n+1)-n}{4 n(n+1)}=\frac{1}{4}\left(\frac{1}{n}-\frac{1}{n+1}\right)$

$S=\frac{1}{4}\left(1-\frac{1}{101}\right)=\frac{1}{4}\left(\frac{100}{101}\right)=\frac{25}{101}$

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