To find the arithmetic sequence consisting of the positive multiples of 4, we first define what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the "common difference."
For the sequence of positive multiples of 4, the terms are formed by multiplying 4 by the positive integers (1, 2, 3, 4, ...).
Let's identify the first few terms of the sequence:
- The first term is [tex]\(4 \times 1 = 4\)[/tex].
- The second term is [tex]\(4 \times 2 = 8\)[/tex].
- The third term is [tex]\(4 \times 3 = 12\)[/tex].
From these terms, we can observe that each term is obtained by adding 4 to the previous term. Therefore, the common difference in this sequence is 4.
We can express the general term of an arithmetic sequence using the formula for the [tex]\(n\)[/tex]-th term, [tex]\(a_n\)[/tex]:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
where:
- [tex]\(a_1\)[/tex] is the first term of the sequence,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the term number.
For our sequence:
- The first term [tex]\(a_1 = 4\)[/tex],
- The common difference [tex]\(d = 4\)[/tex].
Substituting these values into the formula, we get:
[tex]\[ a_n = 4 + (n - 1) \cdot 4 \][/tex]
Simplify the equation:
[tex]\[ a_n = 4 + 4n - 4 \][/tex]
[tex]\[ a_n = 4n \][/tex]
Thus, the formula for the [tex]\(n\)[/tex]-th term of the arithmetic sequence consisting of the positive multiples of 4 is:
[tex]\[ a_n = 4n \][/tex]
Let's verify this formula by finding the 10th term of the sequence:
[tex]\[ a_{10} = 4 \times 10 = 40 \][/tex]
So, the 10th term is 40, which matches our expectations.
Therefore, the formula for the arithmetic sequence of positive multiples of 4 is [tex]\( a_n = 4n \)[/tex].
