# REAL ANALYSIS, Math Induction

1. Aug 29, 2010

### phillyolly

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution

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2. Aug 29, 2010

### 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?

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3. Aug 29, 2010

### phillyolly

One sec, I see my mistake.

4. Aug 29, 2010

### hunt_mat

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

5. Aug 29, 2010

### 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.

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6. Aug 29, 2010

### hunt_mat

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

7. Aug 29, 2010

### phillyolly

I don't understand, sorry.

8. Aug 29, 2010

### hunt_mat

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

9. Aug 29, 2010

### 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.

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10. Aug 29, 2010

### hunt_mat

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

Mat

11. Aug 29, 2010

### phillyolly

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

12. Aug 29, 2010

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.

13. Aug 29, 2010

### phillyolly

Thank you!
It turned out very well.

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