Fibonacci Number plus Constant in terms of Fibonacci Numbers

Theorem
Let $c$ be a number.

Let $\sequence {b_n}$ 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 $\sequence {b_n}$ can be expressed in Fibonacci numbers as:
 * $b_n = c F_{n - 1} + \paren {c + 1} F_n - c$

Proof
The proof proceeds by induction.

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

Basis for the Induction
$\map P 0$ is the case:

Thus $\map P 0$ is seen to hold.

$\map P 1$ is the case:

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P {k - 1}$ and $\map P k$ are true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $b_{k - 1} = c F_{k - 2} + \paren {c + 1} F_{k - 1} - c$
 * $b_k = c F_{k - 1} + \paren {c + 1} F_k - c$

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

Induction Step
This is the induction step:

So $\map P {k - 1} \land \map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

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