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 \( A \) of the 10 arithmetic means between 3 and 7, and finding the value \( B \) of the middle term of the 5 numbers in arithmetic progression (AP) whose sum is 30.

### Step 1: Finding the sum \( A \) of the 10 arithmetic means between 3 and 7

1. **Identify the total number of terms**:
- We have the first term (3), 10 arithmetic means, and the last term (7).
- Therefore, the total number of terms \( n \) is \( 1 + 10 + 1 = 12 \).

2. **Use the formula for the sum of an arithmetic series**:
- The sum \( S_n \) of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (A + L) \]
where \( A \) is the first term, \( L \) is the last term, and \( n \) is the number of terms.

3. **Substituting the values**:
- Here, \( A = 3 \), \( L = 7 \), and \( n = 12 \).
- Thus, we have:
\[ S_{12} = \frac{12}{2} \times (3 + 7) = 6 \times 10 = 60 \]

4. **Finding the sum of the arithmetic means**:
- The sum of the 10 arithmetic means is the total sum minus the first and last terms:
\[ A = S_{12} - (3 + 7) = 60 - 10 = 50 \]

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

1. **Identify the sum of the 5 numbers**:
- The sum of the 5 numbers in AP is given as 30.

2. **Use the formula for the sum of an arithmetic series**:
- For 5 terms, the sum \( S_5 \) can be expressed as:
\[ S_5 = \frac{5}{2} \times (2a + (5-1)d) = 30 \]
where \( a \) is the first term and \( d \) is the common difference.

3. **Rearranging the equation**:
- This simplifies to:
\[ 5 \times (2a + 4d) = 60 \implies 2a + 4d = 12 \]
- Dividing through by 2 gives:
\[ a + 2d = 6 \]

4. **Finding the middle term**:
- The middle term (third term) in the AP is given by:
\[ T_3 = a + 2d \]
- From the previous step, we know \( a + 2d = 6 \), so:
\[ B = 6 \]

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

1. **Add the values of \( A \) and \( B \)**:
- We have \( A = 50 \) and \( B = 6 \).
- Therefore:
\[ A + B = 50 + 6 = 56 \]

### Final Answer:
\[ A + B = 56 \]

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