Summing ______s

Algebra Level 2

Find the value of ∑ n = 1 100 ∫ 0 n x d x . \sum_{n=1} ^{100} \int_0 ^n x\ dx.


The answer is 169175.

Jeff Giff by Jeff Giff , 13, China

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1 solution

Jeff Giff
Jun 10, 2020

Using the basic laws of calculus, ∫ a b f ( x ) d x = F ( b ) − F ( a ) , \int_a ^b f(x)dx = F(b)-F(a), Where F ’ ( x ) = f ( x ) . {F}’(x)=f(x).
So the sum equals to ∑ n = 1 100 1 2 x 2 − 0 = 1 2 ∑ n = 1 100 x 2 . \sum_{n=1} ^{100} \frac{1}{2}x^2-0=\frac{1}{2} \sum _{n=1} ^{100} x^2. Using the ∑ n 2 \sum n^2 formula , the sum is equal to 100 × 101 × 201 6 × 2 = 169175. \frac{100\times101\times201}{6\times2}=169175.

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