How do you prove the 1^2 +2^2+.....+n^2 = n/6 (n+1) (2n+1) by induction?

There are three simple steps to prove anything by induction.

Step 1: When n=1

1²=1

then: 1/6 x 1x (1+1) x (2+1)

=1x2x3/6 =1

Therefore it is true for n=1 

Step 2: Assume it is true for n=k

so, 

1² + 2²+....+k² = k/6 (k+1) (2k+1) 

Step 3: Solve for n= k+1

1²+2²+.....+ k² + (k+1)² = k/6 (k+1) (2k+1) + (k+1)²

= (k+1)/6 (k(2k+1) + 6(k+1) )

= (k+1)/6 (2k²+7k+6)

= (k+1)/6 x (2k+3) x (k+2)

= (k+1)/6 ( (k+1)+1 ) (2(k+1)+1) )

Therefore it is true for n= k+1 

Thererore it must be true for all values.