Consider that 10 arithmetic means are inserted
between 3 and 7 and their sum is a Again consider
that the sum of five numbers in A.P. is 30 and the
value of middle terms is b . Then a + b equals

Options

A. 16
B. 56
C. 46
D. 36

Answer

To solve the problem step by step, we will break it down into two parts: finding the sum of the 10 arithmetic means (denoted as A) and finding the value of the middle term of the 5 numbers in arithmetic progression (denoted as B).

### Step 1: Finding the sum of the 10 arithmetic means (A)

1. **Identify the first and last terms**: The first term (a) is 3 and the last term (l) is 7.
2. **Determine the number of terms**: We have 10 arithmetic means, so the total number of terms (n) is 12 (3 + 10 + 7).
3. **Use the formula for the sum of an arithmetic progression (AP)**:
\[ S_n = \frac{n}{2} \times (a + l) \]
where \( S_n \) is the sum of the first n terms, \( a \) is the first term, and \( l \) is the last term.
4. **Substitute the values**:
\[ S_{12} = \frac{12}{2} \times (3 + 7) = 6 \times 10 = 60 \]
5. **Calculate the sum of the 10 arithmetic means**: Since the sum of the first and last terms (3 and 7) is included in the total sum of 60, we need to subtract these two terms to find the sum of just the 10 means:
\[ \text{Sum of 10 means} = 60 - (3 + 7) = 60 - 10 = 50 \]
Thus, \( A = 50 \).

### Step 2: Finding the middle term of the 5 numbers in AP (B)

1. **Identify the number of terms**: We have 5 terms in this AP.
2. **Let the first term be \( A_1 \) and the common difference be \( D \)**.
3. **The middle term (B) is the third term**:
\[ B = A_1 + 2D \]
4. **Use the formula for the sum of the 5 terms**:
\[ S_5 = \frac{n}{2} \times (2A_1 + (n-1)D) \]
where \( n = 5 \):
\[ 30 = \frac{5}{2} \times (2A_1 + 4D) \]
5. **Multiply both sides by 2**:
\[ 60 = 5(2A_1 + 4D) \]
6. **Divide by 5**:
\[ 12 = 2A_1 + 4D \]
7. **Simplify**:
\[ 6 = A_1 + 2D \]
Thus, we have \( A_1 + 2D = 6 \).
8. **Since \( B = A_1 + 2D \)**, we find that \( B = 6 \).

### Step 3: Calculate \( A + B \)

1. **Add the values of A and B**:
\[ A + B = 50 + 6 = 56 \]

### Final Answer
Thus, the value of \( A + B \) is \( 56 \).

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