Generating Function for Sequence of Powers of Constant/Examples/(b+1)^n - b^n

Example of Generating Function for Sequence of Powers of Constant
Let $b \in \R_{>0}$ be a positive real number.

Let $\sequence {a_n}$ be the sequence defined as:
 * $\forall n \in \Z_{\ge 0}: a_n = \paren {b + 1}^n - b^n$

Then the generating function for $\sequence {a_n}$ is given as:
 * $\map G z = \dfrac z {\paren {1 - b z} \paren {1 - b z - z} }$

Proof
Let $\map {H_1} z$ be the generating function for $\sequence {r_n}$ where:
 * $r_n = b^n$

Then from Generating Function for Sequence of Powers of Constant:
 * $\map {H_1} z = \dfrac 1 {1 - b z}$

Let $\map {H_2} z$ be the generating function for $\sequence {s_n}$ where:
 * $s_n = \paren {b + 1}^n$

Then again from Generating Function for Sequence of Powers of Constant:

From Linear Combination of Generating Functions: