here is a sequence defined by the below recursion formula: $a_n=2a_{n-1}+a_{n-2}$ where $n \in \mathbb{N}$ and $a_0=1,a_1=2$.how to find its closed-form.
How to find its closed-form?
2
$\begingroup$
sequences-and-series
-
1There are a couple of standard approaches. Have you used *generating functions*? – 2012-12-09
2 Answers
3
If we write $E^ra_n=a_{n+r},$
the characteristic/auxiliary equation becomes $E^2-2E-1=0,E=1\pm\sqrt2$
So, the complementary function $a_n=A(1+\sqrt2)^n+B(1-\sqrt2)^n$ where $A,B$ are indeterminate constants to be determined from the initial condition.
$a_0=A+B,$ But $a_0=1$ So, $A+B=1$
and $a_1=A(1+\sqrt2)+B(1-\sqrt2)=2$
Now, solve for $A,B$
1
To put it simple: Use a power trial function $a_n=q^n$ to find a solution disregarding the initial conditions (there will be two solutions $q_1$ and $q_2$ that can work). Next make a linear combination $Aq_1^n+Bq_2^n$ to match the initial conditions. Note that a linear combination is automatically also a solution.