Question

I need help with this

Answer 1

This symbol means that you have to evaluate the generic term $-4n+1$ at n=1,2,3,4,5, and then sum all the terms. Here's the table:

$\begin{array}{c|c}n&-4n+1\\1&-3\\2&-7\\3&-11\\4&-15\\5&-19\end{array}$

So, their sum is

$-3-7-11-15-19=-55$

Alternatively, you manipulate the expression to get

$\displaystyle \sum_{n=1}^5(-4n+1) = \sum_{n=1}^5(-4n) + \sum_{n=1}^5 1 = -4\sum_{n=1}^5 n + \sum_{n=1}^5 1$

For the first sum you can use the formula

$\displaystyle \sum_{n=1}^k n = \dfrac{k(k+1)}{2} \implies \sum_{n=1}^5 n = \dfrac{5\cdot 6}{2} = 15$

The second sum is simply 1 summed 5 times with itself: 1+1+1+1+1=5.

So, we have

$-4\cdot 15 + 5 = -60+5 = -55$