What is the partial sum of ?

Jul 13, 2016

$n \left(n + 1\right) - 20$

Explanation:

${\sum}_{k}^{n} 2 k = 2 {\sum}_{k}^{n} k$

but

$1 + 2 + 3 + 4 + 5 + 6 + \cdots + n = \frac{n \left(n + 1\right)}{2}$

then

${\sum}_{k = m}^{n} k = {\sum}_{k = 1}^{n} k - {\sum}_{k = 1}^{m - 1} k = \frac{n \left(n + 1\right)}{2} - \frac{m \left(m - 1\right)}{2}$

finally

${\sum}_{k = 5}^{n} 2 k = 2 \left(\frac{n \left(n + 1\right)}{2} - \frac{5 \cdot 4}{2}\right) = n \left(n + 1\right) - 20$