https://file.aiquickdraw.com/mathbot/file/1730242839184qwo8c65g.jpg

To solve this problem, let's analyze the information given:

1. **First three terms** of the geometric progression (GP) are \( x + 7 \), \( x + 3 \), and \( x + 1 \).
2. Since these terms are in a geometric progression, the ratio between consecutive terms must be constant.

Let’s denote the common ratio as \( r \). Then, we can set up two equations based on the condition that each term is the previous term multiplied by \( r \).

### Step 1: Set up equations for the common ratio
The common ratio \( r \) can be found by dividing the second term by the first term, and the third term by the second term:
\[
r = \frac{x + 3}{x + 7} = \frac{x + 1}{x + 3}
\]

### Step 2: Cross-multiply to solve for \( x \)
Equate the two expressions for \( r \):
\[
\frac{x + 3}{x + 7} = \frac{x + 1}{x + 3}
\]
Cross-multiplying gives:
\[
(x + 3)^2 = (x + 7)(x + 1)
\]
Expanding both sides:
\[
x^2 + 6x + 9 = x^2 + 8x + 7
\]
Simplify by canceling \( x^2 \) from both sides:
\[
6x + 9 = 8x + 7
\]
Rearrange to solve for \( x \):
\[
2x = 2 \Rightarrow x = 1
\]

### Step 3: Substitute \( x = 1 \) back to find the terms
If \( x = 1 \), then the terms are:
\[
x + 7 = 1 + 7 = 8
\]
\[
x + 3 = 1 + 3 = 4
\]
\[
x + 1 = 1 + 1 = 2
\]
So, the first three terms of the sequence are 8, 4, and 2.

### Step 4: Verify the common ratio
Calculate the common ratio:
\[
r = \frac{4}{8} = \frac{1}{2}
\]
This confirms that the common ratio is indeed \( \frac{1}{2} \), so we have a valid geometric progression.

### Step 5: Find the sum to infinity
The sum to infinity \( S_{\infty} \) of a geometric progression with first term \( a \) and common ratio \( r \) (where \( |r| < 1 \)) is given by:
\[
S_{\infty} = \frac{a}{1 - r}
\]
Here, \( a = 8 \) and \( r = \frac{1}{2} \), so:
\[
S_{\infty} = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16
\]

### Final Answer
The sum to infinity of the series is \( \boxed{16} \).

Would you like further details or have any other questions?

---

### Follow-Up Questions
1. What happens to the sum if the common ratio is negative?
2. Can the sum to infinity be calculated if \( |r| \geq 1 \)?
3. How would we calculate the sum if the series had only a finite number of terms?
4. What if the sequence were arithmetic instead of geometric?
5. How can we determine if a given sequence is geometric?

### Tip
In a geometric progression, the sum to infinity formula only applies if the absolute value of the common ratio is less than 1 (i.e., \( |r| < 1 \)).

The first three terms in a geometric progression are x + 7, x + 3, x + 1. Find the sum to infinity of the series.

Explore how to find the sum to infinity of a geometric progression where the first three terms are given as x + 7, x + 3, and x + 1. This problem covers key concepts such as the common ratio and the sum to infinity formula, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation