Question

Find the sum to n terms of each of the series in Exercises 1 to 7. 3 × 1 2 + 5 × 2 2 + 7 × 3 2 + …

Answer 1

The given series is 1 1×2 + 1 2×3 + 1 3×4 +............... n th trem.

Now, the n th term of the given series is,

a n = 1 n( n+1 )

Using partial fraction method, then n th term can be written as,

a n = 1 n − 1 ( n+1 )

Now, write the first, second, third… n th term of the series.

a 1 = 1 1 − 1 2 a 2 = 1 2 − 1 3 a 3 = 1 3 − 1 4 a n = 1 n − 1 n+1 ,

Adding the above terms, we get

a 1 + a 2 +....+ a n = 1 1 − 1 2 + 1 2 − 1 3 +......+ 1 n − 1 n+1 [ 1 1 + 1 2 + 1 3 +......+ 1 n ]−[ 1 2 + 1 3 + 1 4 +......+ 1 n+1 ] =1− 1 n−1

Solve further.

a 1 + a 2 +....+ a n = n+1−1 n+1 = n n+1

Thus, the sum of series 1 1×2 + 1 2×3 + 1 3×4 +............... n th term is n n+1 .