We were learning about functions in our class, and there came a question: If $f(x+2)-5f(x+1)+6f(x)=0$
$f(0)=2$ and
$f(1)=5$, what is f(x)?
My approach: I found $f(2),f(3)$,..... And they don't seem to follow any general GP or AP pattern.
So I took help of the solution
Solution:
Let a and b be the roots of the equation $x^2-5x+6=0$
$a=2,b=3$
As $a\ne b$,
$f(x)=A.a^x+B.b^x=A.2^x+B.3^x$
From the given values of $f(0)$ and $f(1)$, A and B are found. $f(x)$ comes out to be $2^x +3^x$
My questions:
- Why did they and how could they compare the original equation with a quadratic equation?
2.How f(x) could be explicitly related to the roots of the quadratic equation?
A standard solution for a homogenous linear recurrence relation with constant coefficients such as $f_n+bf_{n-1}+cf_{n-2}=0$ is to find the roots of its characteristic equation $r^2+br+c=0$. If they are distinct, say $r_1$ and $r_2$, then each solution to the relation takes the form $f_n=Ar_1^n+Br_2^n$.