Question

A sequence is defined by `a_n=a_(n-1)+2a_(n-2)`, with `a_1=-3`, `a_2=4`. What is the value of `a_6`?

Answer 1

`a_6=14`. The general formulas `a_n=(-1)^n(10/3)+1/6 2^n` gives

`a_10=10/3+1024/6=174`.

Explanation:

Define the shift operator E by

`Ea_(n-2)=a_(n-1)`

Now, the given equation becomes

`f(E)a_n=(E^2-E-2)a_n=0, n = 1, 2, 3, ...`

The general solution of this difference equation is

`a_n=A(rho_1)^n+ B(rho_2)^n`, where `rho_1 and rho_2` are the

zeros of

`f(rho)=rho^2-rho-2=0`, giving `rho = 1 and 2.`

Now,

`a_n=(-1)^nA+2^nB`

The initial value `a_1=-3` gives

`-A+2B=-3`

and `a_2=4` gives

`A+4B=4`.

Solving,

`A = 10/3 and B =1/6`

It follows that

`a_6=10/3+(2^6)(1/6)=14`.