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Sum of the first n cubes

  • Thread starter ehrenfest
  • Start date
  • #1
2,012
1

Homework Statement


Can someone help me derive the formula for the sum of the first n cubes from the formula for the sum of the first n integers that elucidates the reason why the former is the square of the latter?


Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
rock.freak667
Homework Helper
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Well you could consider the sum from n=1 to N of (n+1)^4-n^4 and then expand using the method of differences
 
  • #3
2,012
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You mean use the fact that it telescopes?

Then you just get N^4-1? How does that help?
 
  • #4
Gib Z
Homework Helper
3,346
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[tex](n+1)^4 - n^4 = 4n^3 + 6n^2 + 4n + 1[/tex] by the Binomial theorem.

Sum BOTH sides, and the LHS should get (N+1)^4 actually. Do you see how the RHS will be expressions in terms of the first N integers to the power of 3, 2, 1 and 0? You already know the cases for 0, 1 and 2, now you can work out 3.
 

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