Final answer:
To find the 7th term in the sequence 12, 15, 18..., one can use the recursive formula an = an-1 + d or the explicit formula an = a1 + (n - 1)d. By applying the common difference of 3, we find that the 7th term is 30.
Step-by-step explanation:
The question is asking to find the 7th term of an arithmetic sequence using both recursive and explicit formulas. An arithmetic sequence is characterized by a constant difference between consecutive terms, which is referred to as the common difference. In the given sequence 12, 15, 18..., the common difference is 3, as each term increases by 3.
Recursive Formula
The recursive formula for an arithmetic sequence can be written as:
an = an-1 + d
Where an is the n-th term, an-1 is the previous term, and d is the common difference.
Starting with the first term a1 = 12 and the common difference d = 3, we can find the 7th term by applying the formula six times, each time substituting the previous term to get the next term.
Explicit Formula
The explicit formula for an arithmetic sequence is given by:
an = a1 + (n - 1)d
Using the first term a1 = 12 and the common difference d = 3, we calculate the 7th term (a7) as follows:
a7 = 12 + (7 - 1) × 3
a7 = 12 + 6 × 3
a7 = 12 + 18
a7 = 30
The 7th term of the sequence is 30.