Sum of an arithmetic progression

Question

Hi I cannot for the life of me remember how to use the arithmetic progression formula can someone help?

I just need to find it for this sequence:

$$5n^2 + (5n^2 - 1) + (5n^2 - 2) + \ldots + (5n^2 - n)$$

Answer 1

The term $5n^2$ appears $(n+1)$ times. (Why).

All that is left to do is add the negative of $1, 2, 3 \cdots n$.

You know the sum for 1 to $n$, and that's it.

Answer 2

We have \begin{align*}5n^2 + (5n^2 - 1) + (5n^2 - 2) + \ldots + (5n^2 - n)&= \overbrace{5n^2+5n^2+\ldots+5n^2}^{(n+1) \text{ times}}-(1+2+\ldots+n)\\ &= 5n^2(n+1)-(1+2+\ldots+n)\end{align*} Now, if $S=1+2+\ldots+n$, then \begin{align*}2S &= (1+2+\ldots+n)+(1+2+\ldots+n)=(n+1)+(n-1+2)+\ldots+\big(n-(n-1)+n\big)\\ &=n(n+1)\end{align*} So that $S=n(n+1)/2$ and thus $$ 5n^2 + (5n^2 - 1) + (5n^2 - 2) + \ldots + (5n^2 - n)= 5n^2(n+1)-\frac{n(n+1)}{2}=\frac{n (n+1) (10 n-1)}{2}$$

Answer 3

As rules of thumb:
An arithmetic sum is the average value of the terms multiplied by the number of terms.
The average of the terms of an arithmetic sequence is the average of the first and last term.

In formulas; if $\left\{ t_k \right\}_{k=0}^n$ is an arithmetic sequence, and $$ S_n=\sum_{k=0}^n t_k $$ then $$ S_n = n\left(\frac{t_0+t_n}{2}\right) $$