ARGHHH....first day of calc1b equivalent...

[SKC]

so I went to my first day of summer classes, without having reviewed anything...integrals, limits flew out of my head...

can anyone help me with this proof?

prove 1^3 + 2^3 + 3^3 + ... + n^3 = [n(n+1) /2]^2

I already showed than when n = 1, the proof is correct, leading to the assumption that it's proved for when n = k...now I need to prove it for when n = k+1 .....any help would be greatly appreciated.

 

[bigredguy]

ooh that hurt my brain, so glad i am done with calc, til college atleast. um you may wanna check out Dr. math. a lot of useful stuff there

 

[SKC]

heh, thx bigredguy

 

[GL]

Here's a hint:

1^3 + 2^3 + 3^3 + ... + n^3 + (n+1)^3 = [n(n+1)/2]^2 + (n+1)^3

 

[SKC]

right...i got to [k(k+1) / 2]^2 + (k+1)^3 ...but then after getting common denominator, my brain stops entirely...


where else but anandtech could one find help within 30 seconds? hehe...thanks to all who have helped. =]

 

[GL]

What you want to prove is that in step (n+1) the left side equals:

[(n+1)[(n+1)+1]/2]^2

so...

Since you've "proven" for n, you can substitute that part into the n+1 step yielding a right side of:

1. [n(n+1)/2]^2 + (n+1)^3

Expand the first square:

2. [n^2(n+1)^2]/2^2 + (n+1)^3

Get common denominator:

3. [n^2(n+1)^2 + (n+1)^3]/4

Factor out a (n+1)^2 term:

4. [(n+1)^2[n^2 + 4(n+1)]]/4

5. [(n+1)^2[n^2 + 4n + 4]]/4

6. [(n+1)^2(n+2)^2]/4 <-- but what do we have here?

We have this:

7. [(n+1)[(n+1)+1]/2]^2

 

[SKC]

thanks a lot GL...sorry, 2 questions =\ ..

<<What you want to prove is that in step (n+1) the left side equals:

[(n+1)[(n+1)+1]/2]^2 >> ...how come?

and in expanding the square, what do you do?

...as my faith in my math skills grows ever smaller.

 

[GoldenBear]

The way we did it was, n=1 is true, and then n=k+1 is true.

So it's not special to the problem, but it's just something that you do..

 

[SKC]

thx GB...hmm, are you at cal?

...that's what we learned too, but I'm not sure of the math involved.

 

[GL]

I'm not sure if I used the correct terminology for expanding the square (I know there is also one involving polynomials. All I mean is that instead of having:

n(n+1)/2 all squared, square the individual parts such as:

n^2(n+1)^2/2^2 since that is effectively what &quot;all squared&quot; means.

 

[SKC]

thanks SO MUCH...I think I understand it now...I reread the expanding the square part; that was a stupid question on my part. Thanks for your patience.

 

[bigredguy]

after this thread i really hope i did good on ap calc test, because i feel really dumb right now.