prove that the sum of the squares of the first n integers is 1/24(2n)(2n+1)(2n+2)?

Question

prove that the sum of the squares of the first n integers is 1/24(2n)(2n+1)(2n+2)

please solve it completely

Answer 1

Can any one proove sum of the squares of the first n integers is 1/24 (2n) (2n+1) (2n+2) step by step instructions pls
Comment by Sunny — April 1, 2007 @ 4:58 am

Sunny: You can use induction to prove your formula. I suggest you do a bit more reading in your text or research online if you’re not sure what that means.
Comment by mrc — April 1, 2007 @ 1:30 pm

Answer 2

Start from the identities: (k+1)^2 - k^2 = 2k + 1 ........(1) (k+1)^3 - k^3 = 3k^2 + 3k + 1 ..........(2) Then: (2) * 2 - (1) * 3 gives: 2[ (k+1)^3 - k^3 ] - 3[ (k+1)^2 - k^2 ] = 6k^2 - 1 .........(3) Note that when summing the left hand side from k = 1 to k = n, all terms in each of the square brackets disappear, except for the first term when k = n and the last term when n = 1. If s is the sum you require, then summing (3) from k = 1 to k = n gives: 6s - 1 = 2[ (n + 1)^3 - 1 ] - 3[ (n + 1)^2 - 1 ] = 2(n + 1)^3 - 3(n +1)^2 + 1 6s = 2(n + 1)^3 - 3(n +1)^2 + 2 = 2n^3 + 6n^2 + 6n + 2 - 3n^2 - 6n + 1 = 2n^3 + 3n^2 + 1 = n(2n + 1)(n + 1) Multiplying by 4 to obtain the rather unusual form you have: 24s = (2n)(2n + 1)(2n + 2) s = (1/24)(2n)(2n + 1)(2n + 2).

Answer 3

THis is correct

n = 1 sum = 1
RHS = 1/24(2)*(3)*(4) = 1
proof by induction

let is be true for n add n+1st term
we need to show for for n+1 terms it is

1/24(2(n+1))(2n+3)(2n+4)
or 1/24(2n+2)(2n+3)(2n+4)
2n(2n+1)(2n+2)/24 + (n+1)^2
= 1/24(2n(2n+1)(2n+2) + 24 (n+1)^2)
= 1/24(((2n+2)(2n(2n+1)+24 (n+1))
= 1/24((2n+2)(4n^2+2n+12n+12)
=1/24(2n+2)(4n^2+14+12)
= 1/24(2n+2)(2n+3)(2n+4)

which is true
hence proved

Answer 4

Let us put the sums
S1=1+2+...+n=n*(n+1)/2
and the unknown
S2=1^2+2^2+...+n^2

We know that
(x+1)^3=x^3+3x^2+3x+1

We put 1,2,....n in the place of x

2^3=1^3+3*1^2+3*1+1
3^3=2^3+3*2^2+3*2+1
........................................
(n+1)^3=n^3+3*n^2+3*n+1

Adding The n Equalities together
and deleting same quantities
on both sides we have

(n+1)^3=3*(1^2+2^2+...+n^2)+
+3*(1+2+...+n)+(n+1)

(n+1)^3=3*S2+3*S1+(n+1)
(n+1)^3=3*S2+3*n*(n+1)/2+(n+1)

3*S2=(n+1)[(n+1)^2-3n/2-(n+1)
3*S2=(n+1)[n^2+2n+1-3n/2-1]
3*S2=(n+1)(n^2+n/2)
6*S2=(n+1)(2n^2+n)
6*S2=(n+1)*n*(2n+1)
S2=n*(n+1)*(2n+1)/6
which is the same result

Answer 5

Sum (X^2) = 2 Integral of Sum of (x)

Sum of X = X(X+1)/2

Integral is X^3/3 + X^2/2 + X

Answer 6

Why don't you pay attention in class and do your own homework?