Which one of the following is a closed form expression for the generating function of the sequence $\{a_n\}$, where $a_n = 2n + 3$ for all $n = 0, 1, 2, \dots$ ?

The ordinary generating function $G(x)$ for the sequence $\{a_n\}$ is defined as:

Splitting the summation:

Using standard generating functions:

• $\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$

• $\sum_{n=0}^{\infty} n x^n = x \frac{d}{dx}\left(\frac{1}{1-x}\right) = \frac{x}{(1-x)^2}$

Substitute these into $G(x)$:

Hence, (D) is the correct option.