# How do you find the partial sum of Sigma (250-8/3i) from i=1 to 60?

Jan 4, 2018

The answer is 10120 (see below).

#### Explanation:

First, the commutative and distributive properties allow us to write:

${\sum}_{i = 1}^{60} \left(250 - \frac{8}{3} i\right) = {\sum}_{i = 1}^{60} 250 - \frac{8}{3} {\sum}_{i = 1}^{60} i$.

Now ${\sum}_{i = 1}^{60} 250$ is just 250 added to itself 60 times. Therefore ${\sum}_{i = 1}^{60} 250 = 60 \cdot 250 = 15000$.

Next, we can use the well-known formula for the sum of the first $n$ integers (see https://brilliant.org/wiki/sum-of-n-n2-or-n3/), which is ${\sum}_{i = 1}^{n} i = 1 + 2 + 3 + \cdots + n = \frac{n \left(n + 1\right)}{2}$, to say

${\sum}_{i = 1}^{60} i = \frac{60 \cdot 61}{2} = 30 \cdot 61 = 1830$.

${\sum}_{i = 1}^{60} 250 - \frac{8}{3} {\sum}_{i = 1}^{60} i$

$= 15000 - \frac{8}{3} \cdot 1830 = 15000 - 4880 = 10120$.

Jan 4, 2018

$10120$

#### Explanation:

${\sum}_{i = 1}^{60} \left(250 - \frac{8}{3} \cdot i\right) = 250 \cdot 60 - \frac{8}{3} \cdot \frac{60 \cdot 61}{2} = 10120$