Rewrite an even of N series as a function of N

[Calpalned]

Homework Statement

If N is even, so that 1+2+3+...+N = (N+1)N/2

Homework Equations

n/a

The Attempt at a Solution

I can easily rewrite the summation as

but I do not know how to justify the question. Thank you.

 

[S.G. Janssens]

I can easily rewrite the summation as

This is not the way to rewrite that summation. It should be $\sum_{x=1}^N{x}$. Your sum just gives $N$.

Can you see a way to group the first and the last term together, then the second and the next to last, etc..?

 

[Calpalned]

I see... If I take N = 4, the it will be 1+2+3+4 = 10
I can group them into (1+4) + (2+3) = 5 + 5. So there are two terms, so 4/2

If N =6 then it will be 1+2+3+4+5+6= 21
and I can group them into (1+6) + (2+5) +(3+4) = 7+7+7, so there are 3 terms, so 6/2

So it looks like the general formula, if N is even, is (N/2)(N+1)
Now I get it!
Thank you!

 

[S.G. Janssens]

Maybe I should have added that it is not necessary to suppose that $N$ is even. The "grouping" trick works generally.

 

[SammyS]

Homework Statement

If N is even, so that 1+2+3+...+N = (N+1)N/2

Homework Equations

n/a

The Attempt at a Solution

I can easily rewrite the summation as View attachment 89954 but I do not know how to justify the question. Thank you.

I see that lately you haven't posted here very much.

As a reminder, please state your complete problem in the body of your thread, no mater what is stated in the title.

It's not at all clear, what you're trying to do here.