If S= 2^2-1 2 + 3^2-2 6 + 4^2-3 12 + upto 10 terms, then S is equal to

Step-by-step solution

The general term T_n of the series is given by T_n = (n+1)^2 - n n(n+1) . Simplifying the numerator: (n+1)^2 - n = n^2 + 2n + 1 - n = n^2 + n + 1 . Thus, T_n = n^2 + n + 1 n(n+1) = n(n+1) + 1 n(n+1) = 1 + 1 n(n+1) . Using partial fractions, 1 n(n+1) = 1 n - 1 n+1 . Therefore, T_n = 1 + 1 n - 1 n+1 . The sum of 10 terms is S = _ n=1 ¹⁰ T_n = _ n=1 ¹⁰ (1 + 1 n - 1 n+1 ) . S = _ n=1 ¹⁰ 1 + _ n=1 ¹⁰ ( 1 n - 1 n+1 ) . The first part is 10 1 = 10 . The second part is a telescoping sum: (1 - 1 2 ) + ( 1 2 - 1 3 ) + + ( 1