WMC 2018 Pursuit - Pack 1, D1

Algebra Level 1

Let 1 6 + 1 12 + 1 20 + 1 30 + 1 42 + 1 56 + 1 72 + 1 90 = a b \dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}+\dfrac{1}{56}+\dfrac{1}{72}+\dfrac{1}{90}=\dfrac{a}{b} , where a a and b b are relatively prime integers. What is a + b a+b ?

WMC Pursuit Questions


The answer is 7.

Julian Yu by Julian Yu , 19, Philippines

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1 solution

The given sum can be expressed as follows:

S = ∑ n = 2 9 1 n ( n + 1 ) = ∑ n = 2 9 ( 1 n − 1 n + 1 ) = 1 2 − 1 10 = 2 5 \begin{aligned} S & = \sum_{n=2}^9 \frac 1{n(n+1)} = \sum_{n=2}^9 \left(\frac 1n - \frac 1{n+1} \right) = \frac 12 - \frac 1{10} = \frac 25\end{aligned}

Therefore, a + b = 2 + 5 = 7 a+b = 2+5 = \boxed 7 .

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