Solving Series Summation Problem: Start & How-To

[tykescar]

It isn't homework, it's in a textbook and I'm having trouble with it.

When r=1, summing to n the series of r^3 = (n^2)/4 (n+1)^2

Show that when r = (n+1), summing to 2n = (n^2)/4 (3n+1)(5n+3)

What order do you start the summation, and how do I begin?

 

[micromass]

I'm sorry, but I just don't understand your question Can you please use mathematical notation, or explain it better? I know it is supposed to be about sums, but I really can't see what sums...

 

[tykescar]

$$\sum_{r=1}^{n}r^3=\frac{n^2}{4}\right\left(n+1)\right^2$$

show that

$$\sum_{r=n+1}^{2n}r^3=\frac{n^2}{4}\right\left(3n+1)\right\left(5n+3)\right$$

I don't understand how to start

 

[micromass]

Aah, thanks for TeXing it, I understand now!

Let's first do the first sum. You should prove such a statements by induction. So, first prove the case n=1. Secondly, assume that n=k has been shown, and prove the case n=k+1.

 

[tiny-tim]

hi tykescar!

just do the sum from 1 to 2n, subtract from it the sum from 1 to n, and do a bit of factoring …

what do you get?