# How do you use the properties of summation to evaluate the sum of Sigma (i-1)^2 from i=1 to 20?

Dec 20, 2016

${\sum}_{i = 1}^{20} {\left(i - 1\right)}^{2} = {\sum}_{i = 1}^{20} \left({i}^{2} - 2 i + 1\right)$

$= {\sum}_{i = 1}^{20} {i}^{2} + {\sum}_{i = 1}^{20} \left(- 2 i\right) + {\sum}_{i = 1}^{20} 1$

$= {\sum}_{i = 1}^{20} {i}^{2} - 2 {\sum}_{i = 1}^{20} i + {\sum}_{i = 1}^{20} 1$

Apply summation formulas to get

$= \frac{\left(20\right) \left(21\right) \left(41\right)}{6} - 2 \frac{\left(20\right) \left(21\right)}{2} + 20$

$= \left(\left(10\right) \left(7\right) \left(41\right)\right) - 2 \left(\left(10\right) \left(21\right)\right) + 20$

$= 2870 - 420 + 20$

$= 2470$