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We are given a geometric progression (GP) where the first term is \( a \) and the common ratio is \( r \). The sum of the first two terms is 12, and the third term is 16.

Let's break this down:

1. The first term is \( a \).
2. The second term is \( ar \).
3. The third term is \( ar^2 \).

### (a) Determine the ratio \( \frac{ar^2}{a + ar} \)

This expression simplifies as:

\[
\frac{ar^2}{a + ar} = \frac{ar^2}{a(1 + r)} = \frac{r^2}{1 + r}
\]

So, the ratio is \( \frac{r^2}{1 + r} \).

### (b) If the first term is larger than the second term, find the value of \( r \).

We are given that the sum of the first two terms is 12, meaning:

\[
a + ar = 12 \quad \text{or} \quad a(1 + r) = 12 \quad \text{so} \quad a = \frac{12}{1 + r}
\]

Also, the third term is 16, so:

\[
ar^2 = 16 \quad \text{so} \quad \frac{12r^2}{1 + r} = 16
\]

Now, we need to solve this equation for \( r \).

Let me calculate that for you.The possible values of \( r \) from solving the equation are \( r = -\frac{2}{3} \) and \( r = 2 \).

However, since the problem states that the first term is larger than the second term (\( a > ar \)), the common ratio \( r \) must be less than 1. Therefore, the valid solution is \( r = -\frac{2}{3} \).

Let me know if you would like more details or have any questions!

### Related questions:
1. How do we derive the general formula for the sum of the first \( n \) terms in a geometric progression?
2. How would the solution change if the first term was smaller than the second term?
3. What happens to the geometric progression if \( r = 1 \) or \( r = -1 \)?
4. How do you find the sum of an infinite geometric series?
5. What are some real-world applications of geometric progressions?

### Tip: 
Always ensure that the common ratio \( r \) satisfies any conditions provided (like \( a > ar \) in this case) to determine which solution is appropriate.

In a geometric progression, the first term is a and the common ratio is r. The sum of the first two terms is 12 and the third term is 16. (a) Determine the ratio ar^2 / (a + ar) (b) If the first term is larger than the second term, find the value of r.

This problem involves solving a geometric progression where the first term is 'a' and the common ratio is 'r'. Learn how to determine the ratio ar^2 / (a + ar) and find the value of 'r' given specific conditions. The sum of the first two terms is 12, and the third term is 16.

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