progressives

Algebra Level pending

2 2 + 2 3 2 + 3 4 2 + 4 5 2 + 5 6 2 + + 100 10 1 2 = ? \large 2^{2} + 2\cdot 3^{2} + 3 \cdot 4^{2} + 4 \cdot 5^{2} + 5 \cdot 6^{2} + \cdots + 100 \cdot 101^{2} = ?


The answer is 26184250.

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1 solution

Francesco Iacca
Feb 19, 2018

Let's call this sum S ( 100 ) S(100) . We can write it in this way:

S ( n ) = i = 1 n i ( i + 1 ) 2 = i = 1 n ( i 3 + 2 i 2 + i ) = i = 1 n i 3 + 2 i = 1 n i 2 + i = 1 n i = n 2 ( n + 1 ) 2 4 + 2 n ( n + 1 ) ( 2 n + 1 ) 6 + n ( n + 1 ) 2 S(n)=\displaystyle \sum_{i=1}^n i(i+1)^2 = \sum_{i=1}^n (i^3+2i^2+i)= \sum_{i=1}^n i^3 + 2\sum_{i=1}^n i^2 + \sum_{i=1}^n i = \frac{n^2(n+1)^2}{4} + 2\frac{n(n+1)(2n+1)}{6} + \frac{n(n+1)}{2}

The answer is S ( 100 ) = 26184250 S(100) = \boxed{26184250}

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