Prove the following by using the principle of mathematical induction for all `n in N`:`1^2+3^2+5^2+dotdotdot+(2n-1)^2=(n(2n-1)(2n+1))/3`

Prove the following by using the principle of mathematical induction f

1.	Prove the following by using the principle of mathematical induction for all `n in N`:`1^2+3^2+5^2+dotdotdot+(2n-1)^2=(n(2n-1)(2n+1))/3`
Answer» `P(k): 1^(2)+3^(2)+5^(2)+…+(2k-1)^(2)=(k(2k-1)(2k+1))/3 `. Now, `{1^(2)+3^(2)+5^(2)+…+(2k-1)^(2)}+{2(k+1)-1}^(2)` `=(k(2k-1)(2k+1))/3 +(2k+1)^(2)= 1/3 *{k(2k-1)(2k+1)+3(2k+1)^(2)}` ` = 1/3 (2k+1){k(2k-1)+3(2k+1)}=1/3 (2k+1)(2k^(2)+5k+3)` ` = 1/3 (k+1)(2k+1)(2k+3)`.