Day 16: Sum and Difference

Expanding the expression to be summed gives $x^2-1$ . Then we can split the sum to give $\sum\limits_{x=1}^{25} x^2 - \sum\limits_{x=1}^{25} 1 = \frac {x (x+1)(2x +1)}{6} - x = 25 \times 13 \times 17 - 25 = 25 \times 220$

Therefore the answer is $\boxed{220}$ .

$\sum\limits_{x=1}^{25}(x+1)(x-1)=\sum\limits_{x=1}^{25}(x^2-1)= \frac{x(x+1)(2x+1)}{6}-1(x)=\frac{25(26)(51)}{6}-1(25)=25(220)$

$\boxed{k=220}$

(x+1)(x-1) = x^2+x-x-1 = x^2-1 sum from 1 to 25 of x^2-1 = sum from 1 to 25 of x^2 -25 sum of x^2 from 1 to 25 = 25(25+1)(2 25+1)/6 - 25 = 25 26*51/6 - 25 = 25(221-1) = 25(220) so k = 220

The given expression = n(n+1)(2n+1)/6 - n , where n = 25 Then the given expression = 25(220) = 25k Then k = 220