Thread: 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.

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.