Check whether the following numbers are in continued proportion: a. 4. 16,64 b.5. 15,50

Answer

To determine whether the given sets of numbers are in continued proportion, we will use the property of continued proportion. According to this property, for three numbers \(a\), \(b\), and \(c\), they are in continued proportion if the square of the middle number \(b\) is equal to the product of the extreme numbers \(a\) and \(c\). In mathematical terms, this can be expressed as:

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

Let's solve the problem step by step for both sets of numbers.

### Part a: Check if 4, 16, 64 are in continued proportion

1. **Identify the numbers**: Here, \(a = 4\), \(b = 16\), and \(c = 64\).

2. **Calculate \(b^2\)**:
\[ b^2 = 16^2 = 256 \]

3. **Calculate \(a \times c\)**:
\[ a \times c = 4 \times 64 = 256 \]

4. **Compare \(b^2\) and \(a \times c\)**:
\[ 256 = 256 \]

Since \(b^2 = a \times c\), we conclude that the numbers 4, 16, and 64 are in continued proportion.

### Part b: Check if 5, 15, 50 are in continued proportion

1. **Identify the numbers**: Here, \(a = 5\), \(b = 15\), and \(c = 50\).

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

3. **Calculate \(a \times c\)**:
\[ a \times c = 5 \times 50 = 250 \]

4. **Compare \(b^2\) and \(a \times c\)**:
\[ 225 \neq 250 \]

Since \(b^2 \neq a \times c\), we conclude that the numbers 5, 15, and 50 are not in continued proportion.

### Summary of Results
- **For 4, 16, 64**: They are in continued proportion.
- **For 5, 15, 50**: They are not in continued proportion.