The square formula of 1 squared Plus to n, and the cubic formula of 1 cubic Plus to n

Proof:

Using formula (n-1) 3 = N3-3N2 +3n-1

S3 = +23 +33 +43 +...+n3

S2 = +22 +32 +42 +...+n2

S1 = 1 +2 +3 +4+...+n

=

S3-3s2+3s1-n = (1-1) 3 + (2-1) (3-1) 3 + (4-1) 3 + ... + (n-1) 3 = S3-n3

=

3S2 = 3s1+n3-n

and s1= N (n+1)/2,

=

S2 = N (n+1) (2n+1)/6

i.e.: +22 +32 +42 +...+n2 = N (n+1) (2n+1)/6 similarly push

Related:

(a+b) ^n
=a^n+c (n,1) a^ (n-1) b+c (n,2) a^ (n-2) b^2+c (n,3) a^ (n-3) b^3+......+c (n,n-2) a^2b^ (n-2) +c (n,n-1) ab^ (n-1) +b^n

Yang Hui Triangle

(a+b) 0-square coefficient 1

(a+b) 1 coefficient of the square 1 1

(a+b) 2-square factor 1 2 1

(a+b) 3-square factor 1 3 3 1

(a+b) 4-square factor 1 4 6 4 1 ...