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., $|r| < 1$).