n aritmetic means are inserted between 7 and 49 and their sum is found to be 364, then n is :

Options

A. 11
B. 12
C. 12
D. 14

Answer

To solve the problem of finding the number of arithmetic means \( n \) inserted between 7 and 49, given that their sum is 364, we can follow these steps:

### Step 1: Understand the Problem
We need to insert \( n \) arithmetic means between the numbers 7 and 49. This creates an arithmetic progression (AP) that starts with 7 and ends with 49.

### Step 2: Identify the Terms of the AP
The first term \( a \) is 7, and the last term \( l \) is 49. The number of terms in this AP will be \( n + 2 \) (including the two endpoints).

### Step 3: Use the Formula for the Sum of an AP
The sum \( S \) of an arithmetic progression can be calculated using the formula:
\[ S = \frac{n}{2} \times (a + l) \]
where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

### Step 4: Substitute the Known Values
In our case, the sum of the arithmetic means is given as 364. Therefore, the total sum of the AP including the means is:
\[ S = 7 + 364 + 49 = 420 \]
Now we can substitute into the sum formula:
\[ 420 = \frac{n + 2}{2} \times (7 + 49) \]

### Step 5: Simplify the Equation
Calculating \( 7 + 49 \):
\[ 7 + 49 = 56 \]
Now the equation becomes:
\[ 420 = \frac{n + 2}{2} \times 56 \]

### Step 6: Clear the Fraction
Multiply both sides by 2 to eliminate the fraction:
\[ 840 = (n + 2) \times 56 \]

### Step 7: Solve for \( n + 2 \)
Now divide both sides by 56:
\[ n + 2 = \frac{840}{56} \]
Calculating \( \frac{840}{56} \):
\[ n + 2 = 15 \]

### Step 8: Solve for \( n \)
Subtract 2 from both sides:
\[ n = 15 - 2 = 13 \]

### Final Answer
Thus, the number of arithmetic means \( n \) is:
\[ \boxed{13} \]