Find a closed form for the generating function for each of these sequences

Question

ind a closed form for the generating function for each of these sequences. (Assume a general form for the terms of the sequence, using the most obvious choice of such a sequence.) a) 0, 0, 3, -3, 3, -3, 3, -3, ......

I don't know how should I solve this kind of problems. can anyone help please

Answer 1

ok so we have following thing

$a_1=0$

$a_2=0$

$a_3=3$

$a_4=-3$

$a_5=3$

$a_6=-3$

we can assume such things that from the beginning

$a_1=0$

$a_2=0$

$a(n+1)=-1^{n}*3$

Answer 2

Hint:

• Generating function for sequence $(-1)^n$ for $n \in \mathbb{N}$ is $$f(x) = \sum_{n=0}^{\infty} (-1)^nx^n = \sum_{n=0}^{\infty}(-x)^n = \frac{1}{1-(-x)} = \frac{1}{1+x}.$$
• If $f$ is a generating function for $(a_0,a_1,a_2,\ldots)$, then the generating function for $(0,0,0,a_0,a_1,\ldots)$ is $$g(x) = x^3f(x).$$

I hope this helps $\ddot\smile$