For all `ninNN`, prove by principle of mathematical induction that,
`1^(2)+3^(2)+5^(2)+ . . .+(2n-1)^(2)=(n)/(3)(4n^(2)-1)`.

For all ninNN , prove by principle of mathematical induction that, 1+3+5+ . . .+(2n-1)=n^(2) .

For all ninNN , prove by principle of mathematical induction that, 2^(2)+5^(2)+8^(2)+ . . . to n terms =(n)/(2)(6n^(2)+3n-1) .

For all ninNN , prove by principle of mathematical induction that, 2^(3n)-1 is divisible by 7.

For all ninNN , prove by principle of mathematical induction that, 15^(2n-1)+1 is divisible by 16.

For all ninNN , prove by principle of mathematical induction that, 1*2+2*3+3*4+ . . . +n(n+1)=(1)/(3)n(n+1)(n+2) .

For all ninNN , prove by principle of mathematical induction that, (1)/(2)+(1)/(4)+(1)/(8)+ . . .+(1)/(2^(n))=1-(1)/(2^(n)) .

For all ninNN , prove by principle of mathematical induction that, sinx+sin3x+ . . .+sin(2n-1)x=(sin^(2)nx)/(sinx)

For all ninNN , prove by principle of mathematical induction that, 2*7^(n)+3*5^(n)-5 is divisible by 24.

For all ninNN , prove by principle of mathematical induction that, 1*1!+2*2!+3*3!+ . . .+n*n ! =(n+1)!-1 .

For all ninNN , prove by principle of mathematical induction that, 1+(1)/(1+2)+(1)/(1+2+3)+ . . .+(1)/(1+2+3+ . . .+n)=(2n)/(n+1) .