The sum of series 1/2+1/6+1/12+1/20........+1/156+1/182

To find the sum 1/2 +1/6+..1/k(k+1).

We know that 1/k(k+1) = 1/k-1/(k+1).

Therefore  the 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/k9k+1) = 1/k-1/(k+1)

Adding the  k terms  we get:

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

Therefore the sum (1/2+1/6+1/12+1/20 ...1/k(k+1) = 1/2-1/(k+1) = (2k+2-2)/2(k+1)= 2k/2(k+1) = k/(k+1)

Here k = 13 

Hence answer. = 

= 13/14

Regards

Arun (askIITians forum expert)