Find the song to Infinity of the geometric progression with the first three terms 0.5, 0.5³ and 0.5⁵.

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?

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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.

Find the sum to infinity of the geometric progression with the first three terms 0.5, 0.5³, and 0.5⁵.

Discover how to calculate the sum to infinity of a geometric progression with terms 0.5, 0.5³, and 0.5⁵. Learn the key concepts, including common ratio and sum to infinity formula. Ideal for students in Grades 9-12 studying sequences and series.

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