Help Please!!! (1 Viewer)

[OH1995]

Hi Guys, i am almost there with this question but can't quite get the last step!!!

Use mathematical induction to prove that for all positive integers n:

1²+3²+5²+...+(2n-1)²= (1/3)n(2n-1)(2n+1)

Thanks!

 

[Drongoski]

Preliminary steps skipped.


Assume true for n = k >= 1








Hence etc etc

 

[fortune_cookie]

Hi Guys, i am almost there with this question but can't quite get the last step!!!

Use mathematical induction to prove that for all positive integers n:

1²+3²+5²+...+(2n-1)²= (1/3)n(2n-1)(2n+1)

Thanks!

Sk= (1/3)k(2k-1)(2k+1)

We must show that Sk+1 = (1/3)(k+1) ( 2(k+1) -1) (2(k+1) +1) = (1/3) (k+1)(2k+1)(2k+3)

Now, Sk+1 = Sk+Tk+1

Sk+1 = [ (1/3)k(2k-1)(2k+1) ] + [ (2(k+1) -1 )^2 ]

= (1/3)k(2k-1)(2k+1) + (2k+1)^2

= (2k+1) [ (1/3)k(2k-1) + (2k+1) ]

= (2k+1) [ (1/3) [2k^2 -k] +2k+1 ]

=(2k+1) [ (2/3)k^2 +(5/3)k +1 ]

= (1/3) (2k+1) [ 2k^2 +5k +3]

= (1/3) (2k+1) (2k^2 +2k +3k +3)

= (1/3) (2k+1) ( 2k(k+1) +3(k+1) )

= (1/3) (2k+1) (2k+3)(k+1)

As required.