For the first \( n \) multiples of a number \( K \), the average is simply:
\[ \text{Average} = K \times \frac{n + 1}{2} \]
Here, \( K = 13 \) and \( n = 5 \):
\[ \text{Average} = 13 \times \frac{5 + 1}{2} = 13 \times 3 = 39 \]
Step 1: Understanding the Concept:
The average (arithmetic mean) of a set of numbers is the sum of the values divided by the total count of numbers in that set.
The multiples of any integer form an Arithmetic Progression (AP) because the difference between any two consecutive terms remains constant. Key Formula or Approach:
For any finite Arithmetic Progression with an odd number of terms, the average is exactly equal to the middle term.
Alternatively, the average of an AP can be calculated as:
\[ \text{Average} = \frac{\text{First Term} + \text{Last Term}}{2} \]
Step 2: Detailed Explanation:
Let us solve using both methods to ensure absolute consistency: Method 1: Direct Sum and Division
1. Find the first five multiples of 13:
- First multiple: \( 13 \times 1 = 13 \)
- Second multiple: \( 13 \times 2 = 26 \)
- Third multiple: \( 13 \times 3 = 39 \)
- Fourth multiple: \( 13 \times 4 = 52 \)
- Fifth multiple: \( 13 \times 5 = 65 \)
2. Sum these five values:
\[ \text{Sum} = 13 + 26 + 39 + 52 + 65 \]
Combine the terms:
\[ \text{Sum} = 13 \times (1 + 2 + 3 + 4 + 5) = 13 \times 15 = 195 \]
3. Divide by the total count of numbers (which is 5):
\[ \text{Average} = \frac{195}{5} = 39 \]
Method 2: Using Properties of an AP
The set \( \{13, 26, 39, 52, 65\} \) is an AP with \( n = 5 \) terms.
The middle term is the third term, which is 39.
Using the boundary formula:
\[ \text{Average} = \frac{\text{First Term} + \text{Last Term}}{2} = \frac{13 + 65}{2} = \frac{78}{2} = 39 \]
Both methods consistently yield 39. Step 3: Final Answer:
The correct option is (B).
