Difference between the square of the sum and the sum of the squares

Algebra Level 3

( ∑ n = 1 100 n ) 2 − ( ∑ n = 1 100 n 2 ) = ? {\left(\displaystyle\sum_{n=1}^{100} n \right)}^{2}-\left(\displaystyle\sum_{n=1}^{100} n^2 \right) = \ ?


The answer is 25164150.

Paulo Guilherme Santos by Paulo Guilherme Santos , 25, Portugal

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1 solution

We have that ( ∑ n = 1 100 n ) 2 − ( ∑ n = 1 100 n 2 ) = ( 100 ⋅ ( 100 + 1 ) 2 ) 2 − ( ∑ n = 1 100 n 2 ) = {\left(\displaystyle\sum_{n=1}^{100} n \right)}^2-\left(\displaystyle\sum_{n=1}^{100} n^2 \right)= { \left( \frac{100 \cdot (100+1)}{2} \right) }^2 -\left(\displaystyle\sum_{n=1}^{100} n^2 \right)=

( 100 ⋅ ( 100 + 1 ) 2 ) 2 − 100 6 ⋅ ( 100 + 1 ) ⋅ ( 2 ⋅ 100 + 1 ) = 25164150 { \left( \frac{100 \cdot (100+1)}{2} \right) }^2- \frac{100}{6}\cdot (100+1) \cdot (2 \cdot 100+1)=25164150

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