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)=(n(n+1))/(2) \end{gathered}`

Finally, using these results, we have:

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