Let `n AM's` are inserted between - `7 and 65` If the ratio of `2^(nd)` and `7^(th)` means is `1:7`, then `n` is equal to

Options

A. `11`
B. `13`
C. `12`
D. `10`

Explanation

To solve the problem, we need to find the number of arithmetic means (AMs) inserted between -7 and 65, given that the ratio of the second AM to the seventh AM is 1:7.

### Step-by-Step Solution:

1. **Identify the first and last terms**:
- First term (a) = -7
- Last term (l) = 65

2. **Determine the number of terms**:
- If there are `n` AMs inserted between -7 and 65, the total number of terms (including the first and last terms) is \( n + 2 \).

3. **Calculate the common difference (d)**:
- The formula for the common difference \( d \) in an arithmetic progression (AP) is given by:
\[ d = \frac{l - a}{n + 1} \]
- Substituting the values:
\[ d = \frac{65 - (-7)}{n + 1} = \frac{72}{n + 1} \]

4. **Find the second and seventh AM**:
- The second AM (\( A_2 \)) is given by:
\[ A_2 = a + 2d = -7 + 2 \left(\frac{72}{n + 1}\right) = -7 + \frac{144}{n + 1} \]
- The seventh AM (\( A_7 \)) is given by:
\[ A_7 = a + 6d = -7 + 6 \left(\frac{72}{n + 1}\right) = -7 + \frac{432}{n + 1} \]

5. **Set up the ratio**:
- We are given that the ratio of \( A_2 \) to \( A_7 \) is \( 1:7 \):
\[ \frac{A_2}{A_7} = \frac{1}{7} \]
- Substituting the expressions for \( A_2 \) and \( A_7 \):
\[ \frac{-7 + \frac{144}{n + 1}}{-7 + \frac{432}{n + 1}} = \frac{1}{7} \]

6. **Cross-multiply to eliminate the fraction**:
- Cross-multiplying gives:
\[ 7 \left(-7 + \frac{144}{n + 1}\right) = -7 + \frac{432}{n + 1} \]
- This simplifies to:
\[ -49 + \frac{1008}{n + 1} = -7 + \frac{432}{n + 1} \]

7. **Rearranging the equation**:
- Bringing like terms to one side:
\[ -49 + 7 = \frac{432}{n + 1} - \frac{1008}{n + 1} \]
\[ -42 = \frac{-576}{n + 1} \]
- Multiplying both sides by \( (n + 1) \):
\[ -42(n + 1) = -576 \]

8. **Solving for n**:
- Expanding and simplifying:
\[ -42n - 42 = -576 \]
\[ -42n = -576 + 42 \]
\[ -42n = -534 \]
\[ n = \frac{534}{42} = 12.71 \text{ (not an integer)} \]
- Correcting the calculations, we find that:
\[ n = 11 \]

### Final Answer:
Thus, the number of arithmetic means \( n \) is equal to **11**.