Question

1^2+2^2+3^2+__+(n-1)^2=​

Answer 1

Answer:

Step-by-step explanation:

have wondered how the closed form for the sum of squares for the first n natural numbers was derived. Given the formula for the sum

1^2+2^2+….+n^2= n(n+1)(2n+1)/6

I learned to prove its correctness using mathematical induction. However, I never understood how the formula was derived in the first place! Knuth writes in The Art of Computer Programming, Volume 1, Third Edition, pp. 32 [Addison-Wesley, June 2009]:

{Most textbooks would simply state those formulas, and prove them by induction. Induction is, of course, a perfectly valid procedure; but it does not give any insight into how on earth a person would ever have dreamed the formula up in the first place, except by some lucky guess.}

While randomly browsing the web, I fortunately stumbled upon a bulletin message that gave a simple derivation of the closed form, as follows:

Let S be the sum of squares of the first n natural numbers, such that

S=1^2+2^2+….+n^2

Our aim is to derive a closed form formula for S in terms of n.