Are the following numbers in continued proportion? a. 9, 15,25 b. 8, 13, 26

Answer

To determine whether the given sets of numbers are in continued proportion, we will follow the property of continued proportion. The property states that for three numbers \(a\), \(b\), and \(c\) to be in continued proportion, the square of the middle term \(b\) must equal the product of the extreme terms \(a\) and \(c\). In mathematical terms, this can be expressed as:

\[ b^2 = a \times c \]

Now, let's solve the given sets of numbers step by step.

### Part a: Checking 9, 15, 25

1. **Identify the numbers**:
- Let \(a = 9\), \(b = 15\), and \(c = 25\).

2. **Calculate \(b^2\)**:
\[ b^2 = 15^2 = 225 \]

3. **Calculate the product of the extreme terms \(a\) and \(c\)**:
\[ a \times c = 9 \times 25 \]
- First, calculate \(9 \times 25\):
\[ 9 \times 25 = 225 \]

4. **Compare \(b^2\) and \(a \times c\)**:
\[ b^2 = 225 \quad \text{and} \quad a \times c = 225 \]
- Since \(225 = 225\), the condition holds.

5. **Conclusion**:
- Therefore, the numbers 9, 15, and 25 are in continued proportion.

### Part b: Checking 8, 13, 26

1. **Identify the numbers**:
- Let \(a = 8\), \(b = 13\), and \(c = 26\).

2. **Calculate \(b^2\)**:
\[ b^2 = 13^2 = 169 \]

3. **Calculate the product of the extreme terms \(a\) and \(c\)**:
\[ a \times c = 8 \times 26 \]
- First, calculate \(8 \times 26\):
\[ 8 \times 26 = 208 \]

4. **Compare \(b^2\) and \(a \times c\)**:
\[ b^2 = 169 \quad \text{and} \quad a \times c = 208 \]
- Since \(169 \neq 208\), the condition does not hold.

5. **Conclusion**:
- Therefore, the numbers 8, 13, and 26 are not in continued proportion.

### Summary of Results:
- For part a (9, 15, 25): **Yes, they are in continued proportion.**
- For part b (8, 13, 26): **No, they are not in continued proportion.**