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Algebra Level 3

1 25 ∑ n = 1 100 n ( n + 1 ) 2 = ? \large \frac{1}{25}\sum_{n=1}^{100}{\frac{n(n+1)}{2}} = \, ?


The answer is 6868.

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Akeel Howell by Akeel Howell , 22, USA

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1 solution

Akeel Howell
Jan 10, 2017

Relevant wiki: Sum of n, n², or n³

1 25 ∑ n = 1 100 n ( n + 1 ) 2 = 1 50 ∑ n = 1 100 ( n 2 + n ) ⟹ 1 50 ⋅ ( ( 100 ) ( 101 ) 2 + ( 100 ) ( 101 ) ( 201 ) 6 ) = 1 50 ⋅ ( 204 ) ( 100 ) ( 101 ) 6 = ( 34 ) ( 100 ) ( 101 ) 50 = 68 × 101 = 6868. \displaystyle{\dfrac{1}{25}\sum_{n=1}^{100}{\dfrac{n(n+1)}{2}}} = \dfrac{1}{50}\sum_{n=1}^{100}{\left(n^2+n\right)} \\ \implies \dfrac{1}{50}\cdot\displaystyle{\left(\frac{(100)(101)}{2}+\dfrac{(100)(101)(201)}{6}\right)} \\ =\dfrac{1}{50}\cdot\dfrac{(204)(100)(101)}{6} = \dfrac{(34)(100)(101)}{50} \\ = 68 \times 101 = \boxed{6868.}

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