Suppose that `n` arithmetic means are inserted between then numbers `7` and `49.` If the sum of these means is `364` then the sum their squares is

Options

A. 103802
B. 11380
C. 11830
D. 18130

Answer

To solve the problem step by step, we need to find the sum of the squares of the arithmetic means inserted between the numbers 7 and 49, given that the sum of these means is 364.

### Step 1: Understand the Problem
We are inserting `n` arithmetic means between the numbers 7 and 49. The first term (a) is 7, and the last term (l) is 49. The sum of the arithmetic means is given as 364.

### Step 2: Set Up the Equation
The total number of terms in the sequence (including the two endpoints) is \( n + 2 \). The sum of an arithmetic series can be calculated using the formula:
\[ S = \frac{n}{2} \times (a + l) \]
Here, \( S \) is the sum of the series, \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

### Step 3: Substitute Known Values
We know:
- \( a = 7 \)
- \( l = 49 \)
- The sum of the means is 364.

Thus, the total sum of the series (including 7 and 49) is:
\[ 7 + 49 + 364 = 420 \]
The number of terms is \( n + 2 \), so we can write:
\[ 420 = \frac{n + 2}{2} \times (7 + 49) \]
\[ 420 = \frac{n + 2}{2} \times 56 \]

### Step 4: Solve for \( n \)
Multiply both sides by 2 to eliminate the fraction:
\[ 840 = (n + 2) \times 56 \]
Now divide both sides by 56:
\[ n + 2 = \frac{840}{56} = 15 \]
Subtract 2 from both sides:
\[ n = 15 - 2 = 13 \]

### Step 5: Find the Common Difference \( d \)
The last term of the arithmetic series can be expressed as:
\[ l = a + nd \]
Substituting the known values:
\[ 49 = 7 + 13d \]
Rearranging gives:
\[ 13d = 49 - 7 = 42 \]
Thus,
\[ d = \frac{42}{13} = 3 \]

### Step 6: Calculate the Sum of Squares
The sum of squares of the arithmetic means can be calculated using the formula:
\[ \text{Sum of squares} = n \cdot a^2 + d^2 \cdot \left(1^2 + 2^2 + \ldots + n^2\right) + 2ad \cdot \left(1 + 2 + \ldots + n\right) \]

1. Calculate \( n \cdot a^2 \):
\[ n \cdot a^2 = 13 \cdot 7^2 = 13 \cdot 49 = 637 \]

2. Calculate \( d^2 \cdot \left(1^2 + 2^2 + \ldots + n^2\right) \):
The formula for the sum of squares of the first \( n \) natural numbers is:
\[ \frac{n(n + 1)(2n + 1)}{6} \]
For \( n = 13 \):
\[ \frac{13 \cdot 14 \cdot 27}{6} = 819 \]
Thus,
\[ d^2 \cdot \left(1^2 + 2^2 + \ldots + n^2\right) = 3^2 \cdot 819 = 9 \cdot 819 = 7371 \]

3. Calculate \( 2ad \cdot \left(1 + 2 + \ldots + n\right) \):
The formula for the sum of the first \( n \) natural numbers is:
\[ \frac{n(n + 1)}{2} \]
For \( n = 13 \):
\[ \frac{13 \cdot 14}{2} = 91 \]
Thus,
\[ 2ad \cdot \left(1 + 2 + \ldots + n\right) = 2 \cdot 7 \cdot 3 \cdot 91 = 3822 \]

### Step 7: Combine All Parts
Now, sum all the parts:
\[ \text{Sum of squares} = 637 + 7371 + 3822 = 11830 \]

### Final Answer
The sum of the squares of the arithmetic means is:
\[ \boxed{11830} \]