Multiples Sum Calculation

The question requires calculating the total sum of the first twelve numbers that are multiples of 6.

Multiples Series Identification

The first 12 multiples of the number 6 are:

• $6 \times 1 = 6$
• $6 \times 2 = 12$
• $6 \times 3 = 18$
• $6 \times 4 = 24$
• $6 \times 5 = 30$
• $6 \times 6 = 36$
• $6 \times 7 = 42$
• $6 \times 8 = 48$
• $6 \times 9 = 54$
• $6 \times 10 = 60$
• $6 \times 11 = 66$
• $6 \times 12 = 72$

This sequence constitutes an arithmetic progression (AP).

Arithmetic Progression Formula

An arithmetic progression is a sequence where the difference between consecutive terms is constant. The sum of the first n terms of an AP is calculated using the formula:

$ S_n = \frac{n}{2}(a + l) $

Where:

• n represents the total number of terms.
• a is the first term in the sequence.
• l is the last term in the sequence.

Formula Application

For this specific problem:

• The number of terms, n = 12
• The first term, a = 6
• The last term, l = 72

Substitute these values into the sum formula:

$ S_{12} = \frac{12}{2}(6 + 72) $

First, calculate the sum inside the parentheses:

$ 6 + 72 = 78 $

Next, perform the multiplication:

$ S_{12} = 6 \times 78 $

$ S_{12} = 468 $

Final Sum

Therefore, the sum of the first 12 multiples of 6 is 468.