Proving Sum of Cubes using Induction | Step-by-Step Guide

[Isaak DeMaio]

1. For all, n is greater/= to 1. 1^3+2^3+...+n^3 = [n(n+1)/2]^2


2. I have to use the K+1 thing. where it would be: K^3+(K+1)^3=[(K+2)(K+3)/2]^2
I may be setting it up wrong, but I used past notes to help and I believe it would look like that.



3. I plugged in 1,2 and 3 into the equations and they check out, so I can move to the next step.
On the left side I get: [K(K+1)/2]^2 + (K+1)^3 = [(K+2)(K+3)/2]^2
I then got a common denominator and for the left side I got [(K^2)(K^2+2K+2)/4] + 4(K^3+3K^2+3K+1)/4 = [(K^2+4K+4)(K^2+6K+9)/4)].

Did I make a mistake at all anywhere in there, cause I kept going but I couldn't get them to equal.

 

[Mark44]

1. For all, n is greater/= to 1. 1^3+2^3+...+n^3 = [n(n+1)/2]^2


2. I have to use the K+1 thing. where it would be: K^3+(K+1)^3=[(K+2)(K+3)/2]^2
I may be setting it up wrong, but I used past notes to help and I believe it would look like that.



3. I plugged in 1,2 and 3 into the equations and they check out, so I can move to the next step.
On the left side I get: [K(K+1)/2]^2 + (K+1)^3 = [(K+2)(K+3)/2]^2
I then got a common denominator and for the left side I got [(K^2)(K^2+2K+2)/4] + 4(K^3+3K^2+3K+1)/4 = [(K^2+4K+4)(K^2+6K+9)/4)].

Did I make a mistake at all anywhere in there, cause I kept going but I couldn't get them to equal.

You need to do three things:
1. Establish a base case.
2. Assume that the proposition is true for n = k. IOW, that
1³ + 2³ + ... + k³ = [k(k+1)/2]^2
This is the induction hypothesis.
3. Show that the induction hypothesis being true implies that the proposition is true for n = k + 1.
IOW, show that 1³ + 2³ + ... + k³ + (k + 1)³ = [(k + 1)(k+2)/2]^2

Start with the left side of the equation above and show that it is equal to [(k + 1)(k+2)/2]^2.

 

[Isaak DeMaio]

You need to do three things:
1. Establish a base case.
2. Assume that the proposition is true for n = k. IOW, that
1³ + 2³ + ... + k³ = [k(k+1)/2]^2
This is the induction hypothesis.
3. Show that the induction hypothesis being true implies that the proposition is true for n = k + 1.
IOW, show that 1³ + 2³ + ... + k³ + (k + 1)³ = [(k + 1)(k+2)/2]^2

Start with the left side of the equation above and show that it is equal to [(k + 1)(k+2)/2]^2.

Can you explain in it and not ask me other questions when I already have a question I can't figure out?

 

[Mark44]

I didn't ask you any questions. I told you what you need to do. What part of what I said don't you understand?