Double sum

Algebra Level 3

∑ m = 1 10 ∑ n = 1 50 m n = ? \large\sum_{m=1}^{10}\sum_{n=1}^{50}{mn}=\, ?


The answer is 70125.

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Akeel Howell by Akeel Howell , 22, USA

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2 solutions

Akeel Howell
Jan 13, 2017

Relevant wiki: Sum of n, n², or n³

∑ m = 1 10 ∑ n = 1 50 m n = ∑ m = 1 10 m ( 50 ) ( 51 ) 2 = ( 25 ) ( 51 ) ∑ m = 1 10 m = ( 25 ) ( 51 ) ( 5 ) ( 11 ) = ( 25 ) ( 51 ) ( 55 ) = 70125. \sum_{m=1}^{10}\sum_{n=1}^{50}{mn} \\ = \sum_{m=1}^{10}{m\dfrac{(50)(51)}{2}} = (25)(51)\sum_{m=1}^{10}{m} \\ = (25)(51)(5)(11) = (25)(51)(55) \\ = \boxed{70125.}

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Y = ∑ m = 1 10 ∑ n = 1 50 m n = ( 1 + 2 + 3 + 4 + . . . . . . . . + 10 ) × ( 1 + 2 + 3 + 4 + . . . . . . + 50 ) = 1 2 ( 10 ) ( 10 + 1 ) × 1 2 ( 50 ) ( 50 + 1 ) Y = \displaystyle \sum_{m=1}^{10} \displaystyle \sum_{n=1}^{50} mn = (1 + 2 + 3+ 4+........+10)\times (1 + 2 + 3 + 4 +......+50) = \dfrac 12 (10)(10+1) \times \dfrac 12 (50)(50+1)

Y = 70125 Y = \boxed{70125}

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