Question

find sum to n terms of series...

please solve it wid full solution.....best will be marked as brainlist..

Answer 1

Hi there!


Query:
To find sum upto n terms, 1/1 × 2 + 1/2 × 3 + 1/3 × 4 +... +1/n(n+1).

Or, 1/2 + 1/6 + 1/12 +... +1/n(n+1).

We know,
1/n(n+1) = 1/n-1/(n+1).

Therefore, term by term we can write as below:

1/2 = (1//1-1/2).

1/6 = (1/2-1/3).

1/12 = (1/3-1/4).

1/20 = (1/4 -1/5).
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(1/n9n+1) = 1/n-1/(n+1)

Adding the n terms  we get:

1/2 + 1/6 + 1/12 + 1/20 +... + 1/(n+1) = 1/2 - 1/(n+1), as all other terms on the right cancel while adding.

Therefore,
The sum (1/2+1/6+1/12+1/20 + ...+ 1/n(n+1) = 1/2-1/(n+1) 
= (2n+2-2)/2(n+1)
= 2n/2(n+1)

= n/(n+1)

Hence, Sum of n terms is n/(n+1)

Cheers!