Combinatorics - Mathematical Induction?

[nintendo424]

Hello, I am having trouble solving this problem. Maybe I'm just overreacting to it. In my two semesters in discrete math/combinatorics, I've never seen a problem like this (with two summations) and been asked to prove it. Can some one help?

$\sum^{n}_{i=1} i^3 = \frac{n^2(n+1)^2}{4} = (\sum^{n}_{i=1} i)^2$

I mean, I know the whole S(n), S(1), S(k), S(k+1) steps, but I'm just unsure of how to write it. The solutions manual for the book skip that problem.

Book: Discrete And Combinatorial Mathematics: An Applied Introduction by Ralph P. Grimaldi, 5th Edition.

 

[micromass]

First of all, this is a textbook problem, so it belongs in the homework forums. I moved it for you

Second, you actually need to show two things:

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

and

$$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$

(and square both sides)

Can you do that?

 

[nintendo424]

Thank you very much! That helped a lot, I just finished my proof. :D That makes sense why you'd have to break it up. I didn't put the relationship between $\sum^{n}_{i=1}i = \frac{n(n+1)}{2}$ and $(\sum^{n}_{i=1}i)^2 = \frac{n^2(n+1)^2}{4}$ together. lol

 

[mathman]

Hello, I am having trouble solving this problem. Maybe I'm just overreacting to it. In my two semesters in discrete math/combinatorics, I've never seen a problem like this (with two summations) and been asked to prove it. Can some one help?

$\sum^{n}_{i=1} i^3 = \frac{n^2(n+1)^2}{4} = (\sum^{n}_{i=1} i)^2$

I mean, I know the whole S(n), S(1), S(k), S(k+1) steps, but I'm just unsure of how to write it. The solutions manual for the book skip that problem.

Book: Discrete And Combinatorial Mathematics: An Applied Introduction by Ralph P. Grimaldi, 5th Edition.

These are two separate problems. ∑i³ is one and ∑i is the other. Have you tried either?

The question belongs in mathematics, not computer science.