Math Induction for Real Analysis Problems: Am I on the Right Track?

[phillyolly]

Homework Statement

The problem and my solution attempt are in the attached file.
Am I doing it right? I didn't write the final answer because it is not what I expected. Just wanted to hear if I made any mistakes. Thank you.

Homework Equations

The Attempt at a Solution

 

[phillyolly]

Here is what I get...I am unable to bring it to an original formula...
Is it right that we cannot prove the expression?

 

[phillyolly]

One sec, I see my mistake.

 

[hunt_mat]

You should be adding (2(k+1)-1)^2=(2k+1)^2 to each side of the equality

 

[phillyolly]

OK, here is my corrected version.
My final answer is bulky. I tried to open brackets but all I get is
(4k^3+4k^2+3k+1)/3.

Please help me from here. Or did I make a mistake before?

 

[hunt_mat]

You forgot to multiply by 3 when you put everything over 3.

 

[phillyolly]

I don't understand, sorry.

 

[hunt_mat]

$$\frac{4k^{3}-k}{3}+(2k+1)^{2}=\frac{4k^{3}-k+3(2k+1)^{2}}{3}$$

 

[phillyolly]

Thank you for the correction.
Where do I go from there? I tried open the brackets. Got
4k^3+12k^2+11k+3. Doesn't look nice.

 

[hunt_mat]

Take out a factor of 4 and ask yourself what the exapnsion of (k+1)^3 is.

Mat

 

[phillyolly]

If I open brackets, I get
4k^3+12k^2+11k+3.
I cannot factor it by 4.

 

[Raskolnikov]

Stop at $$= \frac{4k^3 - k + 3(2k + 1)^2}{3}$$ and expand the numerator completely. You know you want 4(k+1)^3. So do as hunt_mat suggested and expand 4(k+1)^3 as an aside (not in the proof) so you know what it is expanded. Subtract this expanded form from your expanded numerator. It should work.

 

[phillyolly]

Thank you!
It turned out very well.