Consider the series 21, 22, 23, …, k - 1, k where k is a three-digit number. If the A.M. and G.M. of the first and last numbers exist in the given series, then the number of values k can attain is

Options

A. 5
B. 6
C. 2
D. 4

Answer

To solve the problem, we need to analyze the arithmetic mean (A.M.) and geometric mean (G.M.) of the first and last numbers in the series 21, 22, 23, ..., k-1, k, where k is a three-digit number.

### Step 1: Identify the first and last terms of the series
The first term (a) is 21, and the last term (l) is k.

### Step 2: Calculate the Arithmetic Mean (A.M.)
The formula for the A.M. of two numbers a and l is given by:
\[ A.M. = \frac{a + l}{2} = \frac{21 + k}{2} \]

### Step 3: Calculate the Geometric Mean (G.M.)
The formula for the G.M. of two numbers a and l is given by:
\[ G.M. = \sqrt{a \cdot l} = \sqrt{21 \cdot k} \]

### Step 4: Determine the conditions for A.M. and G.M. to be integers
For both A.M. and G.M. to exist in the series, they must be integers.

1. **A.M. is an integer**:
\[ \frac{21 + k}{2} \text{ is an integer} \implies 21 + k \text{ must be even} \]
Since 21 is odd, k must also be odd for their sum to be even.

2. **G.M. is an integer**:
\[ \sqrt{21 \cdot k} \text{ is an integer} \implies 21 \cdot k \text{ must be a perfect square} \]
Since 21 can be factored as \(3 \times 7\), we need \(k\) to be of the form:
\[ k = \frac{n^2}{21} \text{ for some integer } n \]
This means \(n^2\) must be divisible by 21, hence \(n\) must be a multiple of \(3\) and \(7\).

### Step 5: Set up the equation for k
Let \(n = 21m\) for some integer \(m\):
\[ k = \frac{(21m)^2}{21} = 21m^2 \]

### Step 6: Determine the range for k
Since k is a three-digit number, we have:
\[ 100 \leq 21m^2 < 1000 \]
Dividing the entire inequality by 21 gives:
\[ \frac{100}{21} \leq m^2 < \frac{1000}{21} \]
Calculating the bounds:
\[ 4.76 \leq m^2 < 47.62 \]
Taking square roots:
\[ \sqrt{4.76} \leq m < \sqrt{47.62} \]
This simplifies to:
\[ 2.18 \leq m < 6.9 \]
Thus, \(m\) can take the integer values \(3, 4, 5, 6\).

### Step 7: Count the values of k
The possible integer values for \(m\) are \(3, 4, 5, 6\), which gives us a total of:
\[ \text{Number of values for } k = 4 \]

### Final Answer
The number of values k can attain is **4**.

---