Connection between cubed binomial and summation formula proof (for squares)

In summary: So, the equation stated at the beginning of the proof is just a shorthand way of writing down the general form of the summation formula for the sequence of squares.
  • #1
GeoMike
67
0
I was reading through a proof of the summation formula for a sequence of consecutive squares (12 22 + 32 + ... + n2), and the beginning of the proof states that we should take the formula:
(k+1)3 = k3 + 3k2 + 3k + 1
And take "k = 1,2,3,...,n-1, n" to get n formulas which can then be manipulated into the form n(n+1)(2n+1)/6

I can follow the proof without issue, what I'm a little confused about is where "(k+1)3 = k3 + 3k2 + 3k + 1" comes from. It's just given at the beginning of the proof with no logical explanation as to where it came from. I understand that using it allows us to easily get to the final form -- but is there some logical connection between the cubed binomial (and it's expansion) and the sum of the sequence of squares?

Hopefully what I'm asking makes sense...
-GeoMike-
 
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  • #2
I think it should be [tex] (k+1)^{3}-k^{3} = 3k^{2} + 3k +1 [/tex]. And then take [tex] k = 1,2,3, ..., (n-1), n [/tex]
 
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  • #3
courtrigrad said:
I think it should be [tex] (k+1)^{3}-k^{3} = 3k^{2} + 3k +1 [/tex]. And then take [tex] k = 1,2,3, ..., (n-1), n [/tex]

Yes, they subtract k3 from each side in the next step, then take k = 1,2,3,...(n-1),n.

But, what is the process of reasoning that led us to use that equation as a starting point for the proof? For example, if the question had just said "derive a formula for the sum of the sequence of consecutive squares" or "Prove that n(n+1)(2n+1)/6 is a formula for such a sequence" I wouldn't instantly think "Oh yeah, that's easy, I just need to use this cubic equation" -- what process of reasoning would get me there?

-GeoMike-
 
  • #4
I dunno, it's simply mathematical brilliancy. Basically it's intuition plus lots of work on this kind of problems: numerical formulae.

Daniel.
 
  • #5
dextercioby said:
I dunno, it's simply mathematical brilliancy. Basically it's intuition plus lots of work on this kind of problems: numerical formulae.

Daniel.


I can live with that. I just wanted to make sure I wasn't missing something really obvious... :-p

-GeoMike-
 
  • #6
the equation stated to solve the equation (k+1)^3=k^3+3k^2+3k+1 comes from the algebraic proof of 1^2+2^2+3^2+4^2=(1+2+3+4)^3. This also means that 1^2+2^2+3^2+4^2...+n^2=(1+2+3+4...+n)^3. In order to find the summation formula for the square, you need to take the cube.
 

Related to Connection between cubed binomial and summation formula proof (for squares)

What is the connection between the cubed binomial and summation formula proof for squares?

The cubed binomial formula and the summation formula for squares are both mathematical expressions that involve the sum of terms raised to a certain power. They are connected through the process of proving the summation formula for squares using the cubed binomial formula.

What is the cubed binomial formula?

The cubed binomial formula, also known as the binomial theorem, is a formula used to expand a binomial raised to a power. It states that (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.

What is the summation formula for squares?

The summation formula for squares, also known as the sum of squares formula, is a mathematical expression that calculates the sum of the squares of the first n natural numbers. It is given by the formula n(n+1)(2n+1)/6.

How is the cubed binomial formula used to prove the summation formula for squares?

In order to prove the summation formula for squares, we use the cubed binomial formula to expand (n+1)^3 and (n+1)(2n+1)^2. By comparing the expanded forms of these two expressions, we can derive the summation formula for squares.

Why is the connection between the cubed binomial and summation formula proof for squares important?

Understanding the connection between these two formulas allows us to use the cubed binomial formula to derive the summation formula for squares and other related formulas. This can be helpful in solving various mathematical problems and equations.

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