Between the numbers `2` and `20`, `8` means are inserted. Then their sum is

Options

A. `88`
B. `44`
C. `176`
D. None of these

Explanation

To find the sum of the 8 means inserted between the numbers 2 and 20, we can follow these steps:

### Step 1: Identify the first and last terms
The first term \( a_1 \) is 2, and the last term \( a_{10} \) is 20.

### Step 2: Determine the total number of terms
We have 2 endpoints (2 and 20) and we are inserting 8 means between them. Therefore, the total number of terms is:
\[ 10 = 2 + 8 \]

### Step 3: Use the formula for the sum of an arithmetic progression (AP)
In an arithmetic progression, the sum \( S_n \) of the first \( n \) terms can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
where \( n \) is the total number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term.

### Step 4: Substitute the values into the formula
Here, \( n = 10 \), \( a_1 = 2 \), and \( a_{10} = 20 \):
\[ S_{10} = \frac{10}{2} \times (2 + 20) \]

### Step 5: Calculate the sum
Now, calculate the sum:
\[ S_{10} = 5 \times 22 = 110 \]

### Step 6: Find the sum of the means
Since the sum \( S_{10} \) includes the two endpoints (2 and 20), we need to subtract these from the total sum to find the sum of the 8 means:
\[ \text{Sum of means} = S_{10} - (2 + 20) = 110 - 22 = 88 \]

### Conclusion
The sum of the 8 means inserted between 2 and 20 is \( \boxed{88} \).

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