Sum of Arithmetic progression problem

Question

I have following progression:

$2n+(2n-1)+\cdots+n$

which is equivalent to:

$n+(n+1)+\cdots+2n$

The answer says:

You match $2n-i$ with $n+i$ for all $i=0..n$ Therefore you have $(n+1)$ terms of $3n$, so the sum of the $2$ exactly same sequence is $3n(n+1)$ and therefore the sum of 1 sequence is $1.5n(n+1)$

Or you can just apply the formula for the sum of Arithmetic progression, please refer to the wiki page https://en.wikipedia.org/wiki/Arithmetic_progression

Thus, I tried applying sum of arithmetic progression since I need to make it "$1.5n(n+1)$". but when I apply sum of arithmetic progression, it gives me different result like "$\frac n2(2n+(n-1)*1) = 1.5n^2-0.5n$"

How can I get "$1.5n(n+1)$"?

Answer 1

The sum of an artihmetic progression is (first term + last term)*(number of terms)/2.

Here :

• first term=n,
• last term=2n,
• number of terms=n+1

Applying the formula, $Sum=\frac{(n+2n)*(n+1)}{2}$ reaches your desired result...