Question

The summation expression in the following series has an absolute value in it. Expand and evaluate the summation notation. What is the sum of the series?

Answer 1

$|3i-10|=\begin{cases}3i-10&\text{for }3i-10\ge0\\-(3i-10)&\text{for }3i-10$

We have that $3$, which means $|3i-10|$ reduces to $3i-10$ when $i\ge4$, or reduces to $10-3i$ when $i\le3$.

So we can expand the summation as

$\displaystyle\sum_{i=1}^6|3i-10|=\sum_{i=1}^3(10-3i)+\sum_{i=4}^6(3i-10)$

Notice that the 10 contributes a total of 30 from the first sum, and -30 from the second sum, so those terms cancel, leaving us with

$\displaystyle3\left(\sum_{i=4}^6i-\sum_{i=1}^3i\right)=3((4+5+6)-(1+2+3))=3(15-6)=3(9)=27$