S= ⁻¹ ( 1 n^2+n+1 )+ ⁻¹ ( 1 n^2+3 n+3 )+ + ⁻¹ ( 1 1+(n+19)(n+20) ) , then S is equal to :

Step-by-step solution

We know that, array r ⁻¹ 1 1+2 + ⁻¹ 1 1+2 3 + ⁻¹ 1 1+3 4 + + ⁻¹ 1 1+(n-1) n + ⁻¹ 1 1+n(n+1) + + ⁻¹ 1 1+(n+19)(n+20) = ⁻¹ n+19 n+21 ⁻¹ n-1 n+1 + ⁻¹ 1 1+n(n+1) + ⁻¹ 1 1+(n+1)(n+2) + + 1 1+(n+19)(n+20) = ⁻¹ n+19 n+21 ⁻¹ 1 1+n(n+1) + ⁻¹ 1 1+(n+1)(n+2) + + 1 1+(n+19)(n+20) = ⁻¹ n+19 n+21 - ⁻¹ n-1 n+1 array aligned & ⁻¹ ( 1 n^2+n+1 )+ ⁻¹ ( 1 n^2+3 n+3 )+ + & = ⁻¹ ( n+19 n+21 - n-1 n+1 1+ n+19 n+21 n-1 n+1 ) & = ⁻¹ 1 1+(n+19)(n+20) 20 n^2+20 n+1 = S & ⁻¹ ~S = 20 n^2+20 n+1 aligned