Find the generating function of the given sequence: 1, 2, 1, 0, 0*1 point(x + 2)^2x^2 + 1(x + 1)^2x^2 + 2

To find the generating function for the given sequence $1, 2, 1, 0, 0$, we can follow these steps:

1. ### Break Down the Problem

We need to find a generating function that represents the coefficients of the sequence. Since the sequence consists of 5 terms, we will express it in the form of a power series.

2. ### Relevant Concepts

The generating function $G(x)$ of a sequence $a_n$ is given by the formula:

$G(x) = \sum_{n=0}^{\infty} a_n x^n$

For our sequence, we note that $a_0 = 1, a_1 = 2, a_2 = 1, a_3 = 0, a_4 = 0$.

3. ### Analysis and Detail

We can express the generating function as:

$G(x) = 1 \cdot x^0 + 2 \cdot x^1 + 1 \cdot x^2 + 0 \cdot x^3 + 0 \cdot x^4$

Thus,

$G(x) = 1 + 2x + 1x^2$

Collecting these terms, we can write it as:

$G(x) = 1 + 2x + x^2$

4. ### Verify and Summarize

Putting it all together, we see that the generating function accurately represents the coefficients of the sequence:

• Coefficient of $x^0$ is 1
• Coefficient of $x^1$ is 2
• Coefficient of $x^2$ is 1
• Coefficient of $x^3$ and $x^4$ are both 0

Final Answer

Thus, the generating function for the sequence $1, 2, 1, 0, 0$ is:

$G(x) = 1 + 2x + x^2$