JEE Main 2021 (Online) 31st August Morning ShiftQuestion 1
The sum of $10$ terms of the series $\frac{3}{1^{2}\times2^{2}}+\frac{5}{2^{2}\times3^{2}}+\frac{7}{3^{2}\times4^{2}}+\ldots$ is :
Sequence and Series · Telescoping ($V_n$ Method)
Solution
Formulas used. (1) $(n+1)^2-n^2=2n+1$(2) Telescoping: $\sum_{n=1}^{N}(V_n-V_{n+1})=V_1-V_{N+1}$Step 1 - Split. $$\frac{2n+1}{n^2(n+1)^2}=\frac{(n+1)^2-n^2}{n^2(n+1)^2}=\frac1{n^2}-\frac1{(n+1)^2}.$$ Step 2 - Telescope ($n=1$ to $10$): $$1-\frac1{11^2}=\frac{120}{121}.$$ Answer: (C)