Step 1: Identify the First 8 Multiples of 13

The first 8 multiples of 13 are obtained by multiplying 13 by the first 8 positive integers (1 through 8).

• $13 \times 1 = 13$
• $13 \times 2 = 26$
• $13 \times 3 = 39$
• $13 \times 4 = 52$
• $13 \times 5 = 65$
• $13 \times 6 = 78$
• $13 \times 7 = 91$
• $13 \times 8 = 104$

Step 2: Calculate the Sum of the Multiples

The sum of these multiples can be calculated efficiently. Notice that the sum is $13 \times (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)$.

The sum of the first $n$ natural numbers is given by the formula $\frac{n(n+1)}{2}$. For $n=8$, the sum is:

Sum of integers from 1 to 8 = $\frac{8(8+1)}{2} = \frac{8 \times 9}{2} = \frac{72}{2} = 36$.

Therefore, the sum of the first 8 multiples of 13 is:

Sum = $13 \times 36 = 468$.

Alternatively, using the arithmetic progression sum formula $S_n = \frac{n}{2}(a_1 + a_n)$, where $n=8$, $a_1=13$, and $a_8=104$:

Sum = $\frac{8}{2}(13 + 104) = 4 \times 117 = 468$.

Step 3: Calculate the Arithmetic Mean

The arithmetic mean is the sum of the numbers divided by the count of the numbers.

Arithmetic Mean = $\frac{\text{Sum of the first 8 multiples}}{\text{Number of multiples}}$

Arithmetic Mean = $\frac{468}{8}$

Arithmetic Mean = $58.5$.

Conclusion

The arithmetic mean of the first 8 multiples of 13 is 58.5.