The value of the sum of 10 term of the series $S_{10}=\frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+....$ is
Answer & explanation
Correct answer: option 4
10th term of the series = $\frac{1}{20^2-1}$
(Using identity a2-b2 = (a+b)(a-b))
22-1 = 22-12 = (2+1)(2-1) = 3*1
Similarly use the same for other terms
S= 1/(1*3) + 1/(3*5) + 1/(5*7)......
Since they have difference of 2 in denominator terms
Take multiply and divide by 2
S= (1/2) [2/(1*3) + 2/(3*5) + 2/(5*7)......]
S= (1/2) [(3-1)/(1*3) + (5-3)/(3*5) + (7-5)/(5*7)......]
S= (1/2) [(1/1)-(1/3) + (1/3)-(1/5) + (1/5)-(1/7)......+(1/19) - (1/21]
All the +ve and -ve terms are eliminated except 1 and 1/21
S= (1/2) (1- 1/21)
S= 0.5*20/21
The correct answer is Option (4) → $\frac{10}{21}$