# We have f(n) an string;ninNN such that f(n+1)-f(n)=3f(n) and f(0)=-1/2.How to express f(n) according to n?

Jun 28, 2017

${f}_{n} = - {2}^{2 n - 1}$

#### Explanation:

Making

${f}_{n + 1} - 4 {f}_{n} = 0$ with ${f}_{0} = - \frac{1}{2}$

This is a homogeneous linear difference equation with generic solution

${f}_{n} = c {a}^{n}$

Substituting we have

$c {a}^{n + 1} - 4 c {a}^{n} = c \left(a - 4\right) {a}^{n} = 0$ so

${f}_{n} = c {4}^{n}$ but ${f}_{0} = c = - \frac{1}{2}$ so finally

${f}_{n} = - {2}^{2 n - 1}$