# Mathematical Induction

• May 23rd 2012, 06:59 AM
gurnster76
Mathematical Induction
Use mathematical induction to prove that

n
Σ 2/r(r^2-1) = (1/2) - (1/n(n+1)) n= 2,3,4.......
r=2

i have done the basic proof and got to the point

k+1
Σ = (1/2) - (1/k(k+1)) + 2/(k+1)(k+1^2-1)
r=2

I think i need to get to (1/2) - 1/(k+1)(k+2) but im not sure how to get there, any help would be greatly appreciated!
• May 23rd 2012, 07:35 AM
Plato
Re: Mathematical Induction
Quote:

Originally Posted by gurnster76
Use mathematical induction to prove that
n
Σ 2/r(r^2-1) = (1/2) - (1/n(n+1)) n= 2,3,4.......
r=2

i have done the basic proof and got to the point

k+1
Σ = (1/2) - (1/k(k+1)) + 2/(k+1)(k+1^2-1)
r=2

Work with
$\frac{1}{2}-\frac{1}{k(k+1)}+\frac{2}{(k+1)((k+1)^2-1)}=~?$
• May 23rd 2012, 09:50 AM
gurnster76
Re: Mathematical Induction
That's where im stuck.
• May 23rd 2012, 10:04 AM
Plato
Re: Mathematical Induction
Quote:

Originally Posted by gurnster76
That's where im stuck.

$\frac{2}{(k+1)((k+1)^2-1)}=\frac{2}{(k+1)(k^2+2k)}=\frac{2}{k(k+1)(k+2)}$
• May 24th 2012, 12:35 AM
gurnster76
Re: Mathematical Induction
but how do i get from

(1/2) - (1/k(k+1)) + (2/(k+1)(k+2)

to (1/2) - (1/(k+1)(k+2)