Finding Mean of a Set of Abstract Numbers

[cmkluza]

Hello, I'm having a little trouble figuring out the following problem:

Consider the set of number $$a, 2a, 3a, ..., na$$ where $$a$$ and $$n$$ are positive integers.

(i) Show that the expression for the mean of this set is $$\frac{a(n+1)}{2}$$.

So far the only work I've been able to muster up is:

Mean = $$\frac{a+2a+3a+...+na}{n} = \frac{a(1+2+3+...+n)}{n} = a(\frac{1+2+3+...}{n}+1) = \frac{a+2a+3a+...}{n}+a$$

I'm not sure what to do with the indefinitely large sum of numbers that are involved with $$a$$ in this problem, and I'm not really sure how to configure the problem into the simplified expression for mean shown in the problem.

Any help will be greatly appreciated, thanks!

 

[Evgeny.Makarov]

Read about arithmetic progression. In this case, it is sufficient to note that $1+\dots+n=(1+n)+(2+(n-1))+\dots$ where in the second sum the number of terms is $n/2$ and each term equals $n+1$. Consider what $n/2$ means more precisely in this context when $n$ is even and when it is odd.

 

[cmkluza]

Read about arithmetic progression. In this case, it is sufficient to note that $1+\dots+n=(1+n)+(2+(n-1))+\dots$ where in the second sum the number of terms is $n/2$ and each term equals $n+1$. Consider what $n/2$ means more precisely in this context when $n$ is even and when it is odd.

Thanks a bunch for your help. Sorry I wasn't able to reply in a while, but it makes much more sense now. Thanks again!