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((1/2*3) + (1/3*4) +......+(1/9*10)) = 2/5, but how in the shortest possible way(i.e. within a min. or so)?
2 Answers
- RaffaeleLv 78 years ago
1/((n + 1)(n + 2)) can be written as 1/(n + 1) - 1/(n + 2)
therefore for the sum we have
= (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + ... + (1/8 - 1/9) + (1/9 - 1/10) =
= 1/2 + (- 1/3 + 1/3) + ( - 1/4 + 1/4) + (- 1/5 + ... + 1/8 - 1/9 + 1/9) - 1/10
all terms in parenthesis cancel and the sum simplifies to
1/2 - 1/10 = 4/10 = 2/5
this result is very interesting when dealing with very high or not numerical indices, like
((1/2*3) + (1/3*4) +......+(1/(n-1)*n)) which simplifies to 1/2 - 1/n
