Many fractions

Algebra Level 4

1 1 + 2 + 1 1 + 2 + 3 + ⋯ + 1 1 + 2 + 3 + ⋯ + 2016 \large \frac{1}{1+2}+ \frac{1}{1+2+3}+\cdots+ \frac{1}{1+2+3+\dots+2016}

The expression above is in form of a b \dfrac{a}{b} , where a a and b b are coprime positive integers, find a + b a+b .


The answer is 4032.

Tommy Li by Tommy Li , 22, Hong Kong

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1 solution

Chew-Seong Cheong
Nov 21, 2016

S = ∑ n = 2 2016 1 ∑ k = 1 n k = ∑ n = 2 2016 1 n ( n + 1 ) 2 = ∑ n = 2 2016 2 n ( n + 1 ) = 2 ∑ n = 2 2016 ( 1 n − 1 n + 1 ) = 2 ( 1 2 − 1 2017 ) = 2015 2017 \begin{aligned} S & = \sum_{n=2}^{2016} \frac 1{\sum_{k=1}^n k} \\ & = \sum_{n=2}^{2016} \frac 1{\frac {n(n+1)}2} \\ & = \sum_{n=2}^{2016} \frac 2{n(n+1)} \\ & = 2 \sum_{n=2}^{2016} \left( \frac 1n - \frac 1{n+1} \right) \\ & = 2 \left( \frac 1{2} - \frac 1{2017} \right) \\ & = \frac {2015}{2017} \end{aligned}

⟹ a + b = 2015 + 2017 = 4032 \implies a + b = 2015+2017 = \boxed{4032}

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Nice solution! I solved it same as you, sir!

Samuel Job Espartero - 4 years, 1 month ago

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