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<0\end{cases}=\begin{cases}3i-10&\text{for }i\ge\frac{10}3\\\\10-3i&\text{for }i<\frac{10}3\end{cases}`

We have that
`3<\frac{10}3<4`, 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`