Connsecution

Number Theory Level 1

(1/2 x 1/3) + (1/3 x 1/4)...........+(1/99 x 1/100)=

99/100 527/600 101/100 49/100

Rishabh Ghosh by Rishabh Ghosh , 21, USA

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2 solutions

Abdul Ashraf Ali
May 25, 2014

Solved it using Excel Sheet :)

1 Helpful 0 Interesting 0 Brilliant 0 Confused

I solved the problem by seeing that 1/2 breaks down to 1/4+1/4 then 1/4+1/8+1/8... And so on, but by directly comparing each element we see that 1/4>1/6 and 1/8>1/12 and so forth. Using a theorem that states if a i>b i for all i then summation a i> summation b i (i=1 -> n). So if every term in the decomposition of 1/2 is greater than the corresponding term in the series we are looking at then 1/2>the series. Seeing as 49/100 is the only answer less than one half, it must be the correct one.//

Hudson Kirkpatrick - 7 years ago

Note that 1/(n * (n+1)) = (1/n) - (1/(n+1)). We can then see that we are working with a telescoping series, with sum (1/2) - (1/100) = 49/100.

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