Question

Using the principle of mathematical induction, prove that
1² +2²+3² + ... + n² =1/6n(n + 1)(2n + 1) for all n € N.​

Answer 1

Answer:

Let's verify it is correct with n=1:

• 1²=1/6*1*2*3
• 1=1
• confirmed

Let's assume it is correct for n=k

Let's prove this for n=k+1:

• 1/6k(k + 1)(2k + 1)+(k+1)²=
• (k+1)(1/6k(2k+1)+(k+1)=
• 1/6(k+1)(k(2k+1)+6(k+1)=
• 1/6(k+1)(2k²+4k+3k+6)=
• 1/6(k+1)(2k(k+2)+3(k+2))=
• 1/6(k+1)(k+2)(2k+3)=
• 1/6(k+1)((k+1)+1)(2(k+1)+1) ⇔ 1/6n(n+1)(2n+1)

Proved