1- Find the generating functions and the associated sequences of: a. (1 + x) ^ - 1Hint: use the formula: (1 + x) ^ (- n) sum k = 0 to ∞ (- 1) ^ k * C(n + k - 1,k) * x ^ k

Let's begin by solving part **a**, which asks for the generating function and the associated sequence of \( (1 + x)^{-1} \).

### Step 1: Apply the Formula

We are given the general formula for \( (1 + x)^{-n} \) as:
\[
(1 + x)^{-n} = \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} x^k
\]

Here, \( n = 1 \). So, applying this for \( (1 + x)^{-1} \), we have:
\[
(1 + x)^{-1} = \sum_{k=0}^{\infty} (-1)^k \binom{1+k-1}{k} x^k
\]

Simplifying the binomial coefficient \( \binom{1+k-1}{k} = \binom{k}{k} \) (since \( n = 1 \)), and \( \binom{k}{k} = 1 \) for all \( k \).

Thus, the formula becomes:
\[
(1 + x)^{-1} = \sum_{k=0}^{\infty} (-1)^k x^k
\]

### Step 2: Recognize the Generating Function

This is a geometric series with the general form:
\[
\sum_{k=0}^{\infty} (-1)^k x^k = \frac{1}{1 + x}
\]

Therefore, the generating function for \( (1 + x)^{-1} \) is \( \frac{1}{1 + x} \).

### Step 3: Find the Associated Sequence

The associated sequence is the series of coefficients of powers of \( x \) in the expansion of \( (1 + x)^{-1} \). From the series:
\[
\sum_{k=0}^{\infty} (-1)^k x^k
\]
we can see that the sequence is:
\[
1, -1, 1, -1, 1, -1, \dots
\]

### Summary:
- The **generating function** is: \( \frac{1}{1 + x} \).
- The **associated sequence** is: \( 1, -1, 1, -1, 1, -1, \dots \).

Would you like further details on any part of the solution?

Here are 5 related questions to expand on this:
1. What is the generating function for \( (1 + x)^{-2} \)?
2. How would the sequence change if \( (1 + x)^{-3} \) were considered?
3. Can you find the closed form of the generating function for \( (1 - x)^{-1} \)?
4. How does the formula change for \( (1 - x)^n \) for any integer \( n \)?
5. What is the generating function for the binomial series expansion of \( (1 + x)^n \)?

**Tip**: Generating functions are a powerful tool for analyzing sequences in combinatorics and can often simplify complex recurrence relations!

1- Find the generating functions and the associated sequences of: a. (1 + x) ^ -1 Hint: use the formula: (1 + x) ^ (- n) sum k = 0 to ∞ (- 1) ^ k * C(n + k - 1,k) * x ^ k

Explore how to find the generating function and associated sequence for (1 + x)^{-1} using the binomial series expansion. This problem covers generating functions, binomial coefficients, and the geometric series formula, suitable for students in Grades 10-12.

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