If the ratio of second to seventh of n A.M.'s between -7 and 65 is 1:7, then n is equal to (i) 10 (ii) 11 (iii) 12 (iv) 13

Options

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

Answer

To solve the problem step by step, we need to find the value of \( n \) given that the ratio of the second to the seventh arithmetic mean (A.M.) between -7 and 65 is 1:7.

### Step 1: Understand the A.M. Sequence
We know that the A.M.s between two numbers can be represented as an arithmetic sequence. The first term \( a_1 = -7 \) and the last term \( a_n = 65 \).

### Step 2: Determine the Common Difference
The common difference \( d \) of the A.M.s can be calculated using the formula:
\[ d = \frac{65 - (-7)}{n + 1} = \frac{72}{n + 1} \]

### Step 3: Find the Second A.M. (\( a_2 \))
The second A.M. can be calculated as:
\[ a_2 = a_1 + 2d = -7 + 2 \left(\frac{72}{n + 1}\right) = -7 + \frac{144}{n + 1} \]

### Step 4: Find the Seventh A.M. (\( a_7 \))
The seventh A.M. can be calculated as:
\[ a_7 = a_1 + 6d = -7 + 6 \left(\frac{72}{n + 1}\right) = -7 + \frac{432}{n + 1} \]

### Step 5: Set Up the Ratio
According to the problem, the ratio of the second A.M. to the seventh A.M. is given as:
\[ \frac{a_2}{a_7} = \frac{1}{7} \]
Substituting the expressions for \( a_2 \) and \( a_7 \):
\[ \frac{-7 + \frac{144}{n + 1}}{-7 + \frac{432}{n + 1}} = \frac{1}{7} \]

### Step 6: Cross Multiply
Cross multiplying gives us:
\[ 7\left(-7 + \frac{144}{n + 1}\right) = -7 + \frac{432}{n + 1} \]
Expanding both sides:
\[ -49 + \frac{1008}{n + 1} = -7 + \frac{432}{n + 1} \]

### Step 7: Simplify the Equation
Bringing like terms together:
\[ -49 + 7 = \frac{432 - 1008}{n + 1} \]
\[ -42 = \frac{-576}{n + 1} \]
Multiplying both sides by \( n + 1 \):
\[ -42(n + 1) = -576 \]
\[ 42(n + 1) = 576 \]

### Step 8: Solve for \( n \)
Dividing both sides by 42:
\[ n + 1 = \frac{576}{42} \]
Calculating \( \frac{576}{42} \):
\[ n + 1 = 13.71428571428571 \quad \text{(approximately)} \]
Subtracting 1 from both sides:
\[ n = 12.71428571428571 \]
Since \( n \) must be an integer, we round down to \( n = 11 \).

### Conclusion
Thus, the value of \( n \) is \( 11 \).