To find the sum to infinity of a geometric progression (GP), we use the formula:

$S_{\infty} = \frac{a}{1 - r}$

Where:

• $S_{\infty}$ is the sum to infinity,
• $a$ is the first term of the series,
• $r$ is the common ratio of the series, provided that $|r| < 1$ for the sum to infinity to exist.

Step 1: Identify the terms and common ratio

The first three terms of the GP are given as:

• First term $a = 0.5$,
• Second term $0.5^3 = 0.125$,
• Third term $0.5^5 = 0.03125$.

To find the common ratio $r$, we divide the second term by the first term:

$r = \frac{0.5^3}{0.5} = \frac{0.125}{0.5} = 0.25$

So, the common ratio is $r = 0.25$.

Step 2: Apply the sum to infinity formula

Now, substitute $a = 0.5$ and $r = 0.25$ into the formula for the sum to infinity:

$S_{\infty} = \frac{0.5}{1 - 0.25} = \frac{0.5}{0.75} = \frac{1}{1.5} = \frac{2}{3}$

Final Answer:

The sum to infinity of the geometric progression is:

$S_{\infty} = \frac{2}{3}$

Would you like more details on any step, or have any further questions?

---

Here are some related questions:

1. What happens if the common ratio $r$ is negative or larger than 1?
2. Can we find the sum to infinity if $|r| \geq 1$?
3. How do we calculate the sum of the first $n$ terms of a geometric progression?
4. What is the relationship between geometric progressions and exponential functions?
5. How does a geometric series differ from an arithmetic series?

Tip: The sum to infinity formula only works when the absolute value of the common ratio is less than 1, as larger ratios make the series diverge.