# You are subscribed to this thread Proving conjecture for recursive function

 P: 3 Hi, 1. The problem statement, all variables and given/known data 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. Relevant equations anyone care to point me in the right direction? 3. The attempt at a solution not really sure what method to use, induction?
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,318 Since you have two statements: f(2n+1)= 0 and f(2n)= 1/(2n+1), why not try two separate induction proofs?
P: 3
 Quote by HallsofIvy Since you have two statements: f(2n+1)= 0 and f(2n)= 1/(2n+1), why not try two separate induction proofs?
Hey,

"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

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You are subscribed to this thread Proving conjecture for recursive function

 Quote by superdog "f(2n)= 1/(2n+1)" you mean 1/(n+1) right, (not being picky just making sure)
Halls meant exactly what he said: $f(2n)=1/(2n+1)$. You conjecture is that when n is even, $f(n)=1/(n+1)$. Another way of saying n is even is saying that $n=2m$, where $m$ is an integer. Apply this to your conjecture: $f(n)=1/(n+1)\;\rightarrow\; f(2m) = 1/(2m+1)$.

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 $m=1$ as $f(2\cdot1) = 1/(2\cdot1+1) = 1/3$. All that remains is proving the recursive relationship.
 P: 3 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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