Find the recursive formula for the arithmetic sequence 7, 5, 3, 1, −1, . . .
The common difference between consecutive terms in the sequence is -2, since each term is 2 less than the previous term.
To find the recursive formula, we need to find a relationship between the terms in the sequence. Let's consider the n-th term (denoted by an).
For n = 1, the first term in the sequence is 7.
For n = 2, the second term in the sequence is 7 + (-2) = 5.
For n = 3, the third term in the sequence is 5 + (-2) = 3.
For n = 4, the fourth term in the sequence is 3 + (-2) = 1.
And so on.
From this pattern, we can see that each term (except for the first term) is the previous term (an-1) minus 2.
Therefore, the recursive formula for the arithmetic sequence 7, 5, 3, 1, -1, ... is:
an = an-1 - 2, with a1 = 7.