Suppose m arithmeti means are inserted between 1 and 31. If the ratio of the second mean to the mth mean is 1:4, then m is equal to

Options

A. 7
B. 9
C. 11
D. 15

Answer

To solve the problem, we need to find the value of \( m \) when \( m \) arithmetic means are inserted between 1 and 31, and the ratio of the second mean to the \( m \)th mean is \( 1:4 \).

### Step-by-Step Solution:

1. **Understanding the Sequence**:
- We have the first term \( a = 1 \) and the last term \( l = 31 \).
- We are inserting \( m \) arithmetic means between them, resulting in a total of \( m + 2 \) terms.

2. **Finding the Common Difference**:
- The formula for the \( n \)th term of an arithmetic sequence is given by:
\[ a_n = a + (n-1)d \]
- The last term (which is the \( (m+2) \)th term) can be expressed as:
\[ a_{m+2} = 1 + (m + 1)d = 31 \]
- Rearranging this gives:
\[ (m + 1)d = 31 - 1 = 30 \]
- Therefore, we have:
\[ d = \frac{30}{m + 1} \]

3. **Finding the Second and \( m \)th Means**:
- The second mean \( a_2 \) (which is the 2nd term) can be calculated as:
\[ a_2 = a + (2-1)d = 1 + d = 1 + \frac{30}{m + 1} \]
- The \( m \)th mean \( a_m \) (which is the \( m \)th term) can be calculated as:
\[ a_m = a + (m-1)d = 1 + (m-1)d = 1 + (m-1) \cdot \frac{30}{m + 1} \]

4. **Setting Up the Ratio**:
- According to the problem, the ratio of the second mean to the \( m \)th mean is \( 1:4 \):
\[ \frac{a_2}{a_m} = \frac{1}{4} \]
- Substituting the expressions for \( a_2 \) and \( a_m \):
\[ \frac{1 + \frac{30}{m + 1}}{1 + (m-1) \cdot \frac{30}{m + 1}} = \frac{1}{4} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 4 \left( 1 + \frac{30}{m + 1} \right) = 1 + (m-1) \cdot \frac{30}{m + 1} \]
- This simplifies to:
\[ 4 + \frac{120}{m + 1} = 1 + \frac{30(m-1)}{m + 1} \]

6. **Clearing the Denominator**:
- Multiply through by \( m + 1 \) to eliminate the denominator:
\[ 4(m + 1) + 120 = (m + 1) + 30(m - 1) \]
- Expanding both sides:
\[ 4m + 4 + 120 = m + 1 + 30m - 30 \]
- This simplifies to:
\[ 4m + 124 = 31m - 29 \]

7. **Solving for \( m \)**:
- Rearranging gives:
\[ 124 + 29 = 31m - 4m \]
\[ 153 = 27m \]
\[ m = \frac{153}{27} = 5.67 \]
- Since \( m \) must be a whole number, we round to the nearest whole number, which is \( 6 \).

### Final Answer:
Thus, the value of \( m \) is \( 6 \).