Factoring polynomial with two variables

Question

Factoring:

$$f_{n+1}\geq r^{n-2}+r^{n-3}$$ Factoring out a common term of $r^{n-3}$ from line (2), we get: $$f_{n+1}\geq r^{n-3}(r+1)$$

I don't understand how to factor out r^(n-3) to get that result.

Answer 1

$r$ to any power is a common factor of both. I'm sure you are familiar with the following rule for exponents: $r^a \cdot r^b = r^{a+b}$. The other thing you need to know is that $r^0 = 1$. Using this, we can factor out $r^c$ from any expression of the form $r^x + r^y$:

$$ r^x + r^y = r^c ( r^{x-c} + r^{y-c}) $$

In this case, we have $x=n-2$, $y=n-3$, and $c=n-3$, which gives $r^{n-3}(r+r^0) = r^{n-3}(r+1)$.