how would one go about finding the explicit formula for a_n = 2a_(n-1) - a_(n-3) given a_1=2, a_2=3, a_3=5?
Please help, need to get answer quickly! Thanks.
Do you know the characteristic equation method of solving recurrences?
Rewrite the recurrence as a_n - 2a_(n-1) + a_(n-3) = 0
Characteristic equation is r^3 - 2r^2 + 1 = 0
Find all solutions for r.
You should find 3 different roots r_1, r_2, r_3.
Your final formula is a_n = A*r_1^n + B*r_2^n + C*r_3^n
Now use the initial conditions to find A, B, and C.
Yeah, your right. I must be dislexic today.
That's what happens when they don't latexify their mathematics
Your right, it does look like Fibonnaci. I think it is because a_(n-1) - a_(n-3) = a_(n-2) for this set of initial conditions.
Note the recurrence is only Fibonnaci because of the initial conditions provided. Given numbers like 1,2,5 it will no longer be Fibonnaci.
But still the explicit formula for Fibonnaci is quite ugly