Which one of the following is the closed form for the generating function of the sequence $\{ a_n \}_{n \ge 0}$ defined below? $$a_n = \begin{cases} n+1, & n \text{ is odd} \\ 1, & \text{otherwise} \end{cases}$$

The sequence terms are:

• For even $n$: $a_n = 1$

• For odd $n$: $a_n = n + 1 = 1 + n$

We can split $a_n$ as $a_n = 1 + b_n$, where: $$b_n = \begin{cases} n, & n \text{ is odd} \\ 0, & n \text{ is even} \end{cases}$$

The generating function $G(x) = \sum_{n=0}^{\infty} a_n x^n$ can be expressed as:

1. The first sum is the standard geometric series:

2. For the second sum, consider:

Differentiating with respect to $x$:

Multiplying by $x$ gives the desired odd-power series:

Combining both terms:

Thus, option (A) is correct.