Question

I have a mathematical sequence question a picture included this is an arithmetic sequence

Answer 1

Given the sum:

$-4+-3+-2+...+(-5+1n)$

This can be expressed using the sum operator:

$\sum_{k\mathop{=}1}^n(-5+1k)=-4+-3+-2+...+(-5+1n)$

From the left side:

$\sum_{k\mathop{=}1}^n(-5+1k)=\sum_{k\mathop{=}1}^n(-5)+\sum_{k\mathop{=}1}^n(1k)$

It is well known that:

$\begin{gathered} \sum_{k\mathop{=}1}^n(-5)=-5n \\ \\ \sum_{k\mathop{=}1}^n(1k)=\frac{n(n+1)}{2} \end{gathered}$

Finally, using these results, we have:

$\begin{gathered} \sum_{k\mathop{=}1}^n(-5+1k)=-5n+\frac{n(n+1)}{2}=\frac{-10n+n^2+n}{2} \\ \\ \therefore\sum_{k\mathop{=}1}^n(-5+1k)=\frac{n(n-9)}{2} \end{gathered}$