Question

Prove by Mathematical Induction:

2^{2 +} 4² + 6² + 8² + ..... + n² = n (n+1) (n+2)/6

Answer 1

Case n = 1:
----------------
LHS = 4
RHS = 1

LHS!=RHS
So, given equation is false

===================================================
This is the other proof related to your question
1^2 + 2^2 + 3^3 + ... + n^2 = n(n+1)(n+2)/6
 
Case n = k + 1
--------------------
LHS:
= 2^2 + 3^3 + ... + k^2 + (k+1)^2
= (k(k+1)(k+2))/6 + (k+1)^2
= (k+1) (k(k+2)/6 + (k+1))
= (k+1) (k(k+2) + 6(k+1))/6
= (k+1) (k^2 + 2k + 6k + 6)/6
= (k+1) (k^2 + 2k + 6k + 6)/6
= (k+1) (k^2 + 8k + 6)/6
= (k+1)(k+2)(k+3)/6

RHS:
= (k+1)(k+1+1)(k+1+2)/6
= (k+1)(k+2)(k+3)/6


LHS = RHS
Hence proved