Determining Formulas for Induction

For the following function f with domain N determine a formula for f(n) and use mathematical induction to prove your conclusion:

For a1,a2 in R, arbitrary, let f(1)=a1 and f(2)=a2. For n>2 f(n)=- ^{f(n-2)}/_{n(n-1)}

I calculated the first few values as :

f(3)=-^{a1}/_{6 }f(4)=-^{a2}/12

f(5)=^{a1}/120

f(6)=^{a2}/360

I was going to define this piecewise, since a1 goes with the odd n and a2 is paired with even ones. I've come up with

^{a1}/_{n(n-1)(n-2)(n-3) }for even n

^{a2}/_{n(n-1)(n-2)(n-3) }for odd n

I'm not really sure how to address the changing sign however. I feel like I should use -1 to some power, but I'm not sure how to approach it.

Re: Determining Formulas for Induction

Hey renolovexoxo.

In terms of the changing sign parameter, you might want consider (-1)^([n/2]) where [n/2] is the floor function:

Floor and ceiling functions - Wikipedia, the free encyclopedia

The floor function will generate for n = 0,1,2,3 the values of 0,0,1,1 which gives the kind of behaviour you are looking for.

Re: Determining Formulas for Induction

$\displaystyle f(n)=-\left(\frac{1}{n}-\frac{1}{n-1}\right) f(n-2)$

$\displaystyle f(n) = \frac{\left(a_2-\frac{a_1}{2}\right) (-1)^n+\frac{1}{2} \left(a_1+2 a_2\right)}{\Gamma (n+1)}$