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We are given that the second and third terms of a geometric progression (GP) are 10 and 8, respectively. We need to find the sum to infinity of this progression.

### Step 1: Express the second and third terms in terms of the first term and common ratio
Let the first term of the GP be \( a \) and the common ratio be \( r \). 

- The second term is \( ar \), and we know \( ar = 10 \).
- The third term is \( ar^2 \), and we know \( ar^2 = 8 \).

So, we have the system of equations:
\[
ar = 10 \tag{1}
\]
\[
ar^2 = 8 \tag{2}
\]

### Step 2: Solve for the common ratio \( r \)
From equation (1), solve for \( a \):
\[
a = \frac{10}{r}
\]
Substitute this into equation (2):
\[
\frac{10}{r} \cdot r^2 = 8
\]
Simplifying:
\[
10r = 8
\]
\[
r = \frac{8}{10} = 0.8
\]

### Step 3: Solve for the first term \( a \)
Substitute \( r = 0.8 \) into equation (1):
\[
a \cdot 0.8 = 10
\]
\[
a = \frac{10}{0.8} = 12.5
\]

### Step 4: Find the sum to infinity
The formula for the sum to infinity of a GP is:
\[
S_{\infty} = \frac{a}{1 - r}
\]
Substituting the values of \( a = 12.5 \) and \( r = 0.8 \):
\[
S_{\infty} = \frac{12.5}{1 - 0.8} = \frac{12.5}{0.2} = 62.5
\]

Thus, the sum to infinity of the geometric progression is **62.5**.

---

Would you like more details or have any questions? Here are five related questions to deepen your understanding:

1. What happens if the common ratio \( r \) is greater than 1? Does the sum to infinity still exist?
2. How would the process change if the second and third terms were negative values?
3. Can you derive a general formula for the \( n \)-th term of a geometric progression?
4. How is the sum to infinity formula derived mathematically?
5. What real-world applications exist for geometric progressions and their sums?

**Tip:** Always ensure that the common ratio \( r \) in a geometric series lies between \(-1\) and \(1\) for the sum to infinity to converge.

The second and third terms of a geometric progression are 10 and 8 respectively. Find the sum to infinity.

Find the sum to infinity of a geometric progression where the second term is 10 and the third term is 8. This problem involves solving for the first term and common ratio using algebraic methods and then applying the formula for the sum to infinity of a GP.

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