SOLUTION: Prove by mathematical induction 1^2+3^2+5^2+.......upto n terms = n(4n^2-1)/3

Question 1051728: Prove by mathematical induction
1^2+3^2+5^2+.......upto n terms = n(4n^2-1)/3
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
1) 1^2+3^2+5^2+.......up to (2n-1)^2 = n(4n^2-1)/3 = n(2n-1)(2n+1)/3
:
show that 1) is true for n = 1, that is
:
1^2 = 1(3) / 3 = 1
:
now assume 1) is true for n, then show it is true for n+1
:
n(2n-1)(2n+1)/3 + (2n+1)^2 = (2n+1)(2n^2 -n +6n +3) / 3 =
:
*********************************************************
(2n+1)(2n^2 +5n +3) /3 = (2n+1)(2n+3)(n+1) / 3
:
note that if we substitute (n+1) for n in 1) we get
:
(n+1)(2(n+1)-1)(2(n+1)+1)/3 = (n+1)(2n+1)(2n+3)/3
*********************************************************
:
the statement has been proven with induction
:
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