Fibonacci Number plus Constant in terms of Fibonacci Numbers

Theorem
Let $c$ be a number.

Let $\left\langle{b_n}\right\rangle$ be the sequence defined as:
 * $b_n = \begin{cases}

0 & : n = 0 \\ 1 & : n = 1 \\ b_{n - 2} + b_{n - 1} + c & : n > 1 \end{cases}$

Then $\left\langle{b_n}\right\rangle$ can be expressed in Fibonacci numbers as:
 * $b_n = c F_{n - 1} + \left({c + 1}\right) F_n - c$

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:
 * $b_n = c F_{n - 1} + \left({c + 1}\right) F_n - c$

$P \left({0}\right)$ is the case:

Thus $P \left({0}\right)$ is seen to hold.

Basis for the Induction
$P \left({1}\right)$ is the case:

Thus $P \left({1}\right)$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k + 1}\right)$ is true.

So this is the induction hypothesis:
 * $b_k = c F_{k - 1} + \left({c + 1}\right) F_k - c$

from which it is to be shown that:
 * $b_{k + 1} = c F_k + \left({c + 1}\right) F_{k + 1} - c$

Induction Step
This is the induction step:

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_{\ge 0}: b_n = c F_{n - 1} + \left({c + 1}\right) F_n - c$