Sum of n^3 = (sum of n)^2 induction proof

[cnwilson2010]

Homework Statement

Prove:
1^3 + 2^3 +...+n^3=(1+2+...+n)^2
n=Natural number

Homework Equations

The Attempt at a Solution

Using induction -
n=1 obvious

Assume for n=k equation is true.

Show for k+1.

I have that the right side prior to k+1 is (k^2(k+1)^2)/4
After k+1 I have (k^4+6k^3+13k^2+12k+4)/4,
but I can't figure out the trick to get the form above equal to the left +(k+1)^3.

Any ideas?
Thanks.

 

[tiny-tim]

welcome to pf!

hi cnwilson2010! welcome to pf!

(try using the X² icon just above the Reply box )

erm … a² - b² = … ? ​

 

[Dick]

Homework Statement

Prove:
1^3 + 2^3 +...+n^3=(1+2+...+n)^2
n=Natural number

Homework Equations

The Attempt at a Solution

Using induction -
n=1 obvious

Assume for n=k equation is true.

Show for k+1.

I have that the right side prior to k+1 is (k^2(k+1)^2)/4
After k+1 I have (k^4+6k^3+13k^2+12k+4)/4,
but I can't figure out the trick to get the form above equal to the left +(k+1)^3.

Any ideas?
Thanks.

On the left side you've added (k+1)^3. On the right side you've added the results after k+1 minus the results after k. The thing you want to prove equal to (k+1)^3 is (k+1)^2*(k+2)^2/4-k^2*(k+1)^2/4.

 

[cnwilson2010]

Thank you for your replies they were very helpful. Now I'm working on n²<=2ⁿ.

For what values of Natural numbers is this statement true and prove by induction. Obviously, 1 and 2 are true, now I'm thinking proof by contradiction and the idea of the existence of a least member of the set. Does this sound reasonable or way too much work for the results?

 

[hunt_mat]

Seems begging for proof by induction, you will have to show that 2k+1&lt;k^{2} at some point.

Or failing that, you could note that the two sequences a_{n}=n^{2},b_{n}=2^{n} are both monotonically increasing sequences and that one increases faster that the other.