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You are subscribed to this thread Proving conjecture for recursive function

  • Thread starter superdog
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Hi,

1. Homework Statement

Just having some troubles with a proof i have been asked to do, (sorry for not knowing the math code)

basically, f(1)=0, f(2)=1/3 and f(n)= ((n-1)/(n+1))*f(n-2)

and i've come up with the conjecture that f(n) = 0 when n is odd, and = 1/(n+1) when n is even.

and i have to prove my conjecture, this is where i'm stuck,

2. Homework Equations

anyone care to point me in the right direction?

3. The Attempt at a Solution

not really sure what method to use, induction?
 

Answers and Replies

HallsofIvy
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Since you have two statements:
f(2n+1)= 0 and f(2n)= 1/(2n+1), why not try two separate induction proofs?
 
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Since you have two statements:
f(2n+1)= 0 and f(2n)= 1/(2n+1), why not try two separate induction proofs?
Hey,

thanks for your help,

"f(2n)= 1/(2n+1)" you mean 1/(n+1) right, (not being picky just making sure)

yeah, i kinda what your saying, ie: 2n is a generator for all evens and 2n+1 is for odds, however, i'm not really confident on how to actually go through and do a induction proof on either, the recursive function is knda scaring me at the moment
 
D H
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"f(2n)= 1/(2n+1)" you mean 1/(n+1) right, (not being picky just making sure)
Halls meant exactly what he said: [itex]f(2n)=1/(2n+1)[/itex]. You conjecture is that when n is even, [itex]f(n)=1/(n+1)[/itex]. Another way of saying n is even is saying that [itex]n=2m[/itex], where [itex]m[/itex] is an integer. Apply this to your conjecture: [itex]f(n)=1/(n+1)\;\rightarrow\; f(2m) = 1/(2m+1)[/itex].

To prove some conjecture by induction, you need to show two things:
  • That the conjecture is true for some base case and
  • That if the conjecture is true for some m then it is also true for m+1.

The conjecture is obviously true for [itex]m=1[/itex] as [itex]f(2\cdot1) = 1/(2\cdot1+1) = 1/3[/itex]. All that remains is proving the recursive relationship.
 
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Ahhhh, ok, thanks so much for the help! its all clicking into place now. especially after walking away from the bloody thing for a bit :P

thanks again
Sam
 

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