How Do You Derive the Formula for the Sum of Consecutive Squares?

[sitedesigner]

ok, i need to derive a formula that will add the consecutive squares of n numbers.

for example $$1^2 + 2^2 + 3^2 + ... + (n-2)^2 + (n-1)^2 + (n)^2$$

I have worked on this problem for quite some time and haven't been able to come up with anything.

I do know that the sum of consecutive numbers starting at one is

$$\frac{n}{2} (n+1)$$

A very detailed explanation would be excellent as that's what my professor wants.

 

[iNCREDiBLE]

$$S(n) = \frac{n(n+1)(2n+1)}{6}$$.

You should be able to prove it by induction.

 

[sitedesigner]

what's induction?

can you explain how you came to the answer?

 

[daveed]

inductions like this...
prove it works for 1
assume it works for n
and prove it works for n+1

 

[pathre]

(n-1)^2-n^2-1/-2=t
s=summation of t
we will get s

 

[iNCREDiBLE]

Dr. Math has answered a lot of questions concerning the sum of consecutive squares here. He explains that there are several ways to derive the formula.

 

[sitedesigner]

ok, so i understand what inductions are, but can you explain how you got to the proof for the sum of the sequence of $$n^2$$

*** edit *** i just posted the above before reading the previous 3 posts. i'll go ahead and read dr math's explanation and then come back to this :)

 

[sitedesigner]

inductions like this...
prove it works for 1
assume it works for n
and prove it works for n+1

i need to derive the forumula... not prove it :)

 

[amcavoy]

i need to derive the forumula... not prove it :)

Sequence of Differences. Search here or Dr. Math, there are explanations at both places.