Recursively defines function

Question

Find $f(2),f(3),f(4)$, and $f(5)$ if $f$ is defined recursively by $f(0)=-1$, $f(1)=2$, and for $n=1,2,\ldots$

$$f(n+1)= 3f(n)^2 - 4f(n-1)^2$$

Answer 1

This is just a matter of substituting values into the recurrence. For example, when $n=1$ the recurrence becomes

$$\begin{align*} f(1+1)&=3f(1)^2-4f(1-1)^2\\ &=3\cdot2^2-4(-1)^2\\ &=12-4\\ &=8\;, \end{align*}$$

so $f(2)=8$. Now you should be able to go ahead and calculate the other three requested values; just take them one at a time, because you need $f(3)$, for instance, in order to calculate $f(4)$.