Let 3 + 4 + 8 + 9 + 13 + 14 + 18 +……….40 terms = S. If S = (102)m then m =

Options

A. (a)5
B. (b)20
C. (c)25
D. (d)10

Answer

To solve the problem, we need to find the sum \( S \) of the first 40 terms of the series given by \( 3, 4, 8, 9, 13, 14, 18, \ldots \) and express it in the form \( S = (102)^m \) to find the value of \( m \).

### Step-by-Step Solution:

1. **Identify the Series**:
The series is \( 3, 4, 8, 9, 13, 14, 18, \ldots \).

2. **Group the Terms**:
We can group the terms in pairs:
- \( (3 + 4) \)
- \( (8 + 9) \)
- \( (13 + 14) \)
- \( (18 + 19) \)
- Continue this pattern until we reach 40 terms.

3. **Calculate the Pairs**:
- The first pair: \( 3 + 4 = 7 \)
- The second pair: \( 8 + 9 = 17 \)
- The third pair: \( 13 + 14 = 27 \)
- The fourth pair: \( 18 + 19 = 37 \)
- Continue this until the 20th pair.

4. **Determine the Number of Pairs**:
Since we have 40 terms and we are pairing them, we will have \( 20 \) pairs.

5. **Identify the New Series**:
The new series formed by the sums of the pairs is:
- \( 7, 17, 27, 37, \ldots \)

6. **Recognize the New Series as an Arithmetic Progression (AP)**:
- First term \( a = 7 \)
- Common difference \( d = 10 \)
- Number of terms \( n = 20 \)

7. **Use the Formula for the Sum of an AP**:
The sum \( S_n \) of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Substituting the values:
\[ S = \frac{20}{2} \times (2 \times 7 + (20 - 1) \times 10) \]
\[ S = 10 \times (14 + 190) = 10 \times 204 = 2040 \]

8. **Express \( S \) in the Required Form**:
We need to express \( S = 2040 \) in the form \( S = (102)^m \).
\[ 2040 = 102 \times 20 \]
Thus, we can write:
\[ S = 102^1 \times 20 \]
Therefore, comparing with \( S = (102)^m \), we have \( m = 20 \).

### Final Answer:
\[ m = 20 \]