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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).
..............................
..................................
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)
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