Discussion
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 \(n = 0, 1, 2, \ldots\) ?
Answer: D
The generating function is \(\frac{3-x}{(1-x)^{2}}\).
Split the sequence into its two parts and use the standard series \(\sum x^{n} = \frac{1}{1-x}\) and \(\sum n x^{n} = \frac{x}{(1-x)^{2}}\):
\[ \begin{aligned} \sum (2n + 3)x^{n} &= \frac{2x}{(1-x)^{2}} + \frac{3}{1-x} \\ &= \frac{2x + 3(1-x)}{(1-x)^{2}} = \frac{3-x}{(1-x)^{2}} \end{aligned} \]
Check: the expansion starts \(3 + 5x + 7x^{2}\), matching \(a_{0} = 3\), \(a_{1} = 5\). \(\frac{3}{(1-x)^{2}}\) starts 3 + 6x, \(\frac{3x}{(1-x)^{2}}\) starts with 0, and \(\frac{2-x}{(1-x)^{2}}\) starts with 2, so none of them fits.
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