Five distinct 2-digit numbes are in a geometric progression. Find the middle term.

Answer

To find the middle term of five distinct two-digit numbers that are in a geometric progression (GP), we can follow these steps:

### Step 1: Understand the properties of a geometric progression
In a geometric progression, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For five terms, we can denote them as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- Fifth term: \( ar^4 \)

### Step 2: Identify the range for two-digit numbers
The smallest two-digit number is 10, and the largest is 99. Therefore, we need to ensure that all terms \( a, ar, ar^2, ar^3, ar^4 \) fall within this range.

### Step 3: Set up inequalities
To ensure all terms are two-digit numbers:
1. \( a \geq 10 \)
2. \( ar^4 \leq 99 \)

### Step 4: Find suitable values for \( a \) and \( r \)
Since we need five distinct two-digit numbers, we can try different integer values for \( a \) and rational values for \( r \). A common approach is to start with \( a = 16 \) (the smallest two-digit number that allows for a reasonable common ratio).

### Step 5: Choose a common ratio
Let's choose \( r = \frac{3}{2} \) (1.5) because it keeps the terms within the two-digit range.

### Step 6: Calculate the terms of the GP
Using \( a = 16 \) and \( r = \frac{3}{2} \):
1. First term: \( a = 16 \)
2. Second term: \( ar = 16 \times \frac{3}{2} = 24 \)
3. Third term: \( ar^2 = 16 \times \left(\frac{3}{2}\right)^2 = 16 \times \frac{9}{4} = 36 \)
4. Fourth term: \( ar^3 = 16 \times \left(\frac{3}{2}\right)^3 = 16 \times \frac{27}{8} = 54 \)
5. Fifth term: \( ar^4 = 16 \times \left(\frac{3}{2}\right)^4 = 16 \times \frac{81}{16} = 81 \)

### Step 7: List the terms
The five terms are:
- 16, 24, 36, 54, 81

### Step 8: Identify the middle term
The middle term of the five terms is the third term:
- Middle term = 36

### Conclusion
Thus, the middle term of the five distinct two-digit numbers in geometric progression is **36**.

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