An arithmetic sequence is represented by the explicit formula A(n) = 2 − 5(n − 1). What is the recursive formula?
A. A(n) = A(n − 1) + 5
B. A(n) = A(n − 1) + 2
C. A(n) = A(n − 1) − 5
D. A(n) = A(n − 1) − 2
since you just keep subtracting 5,
A(1) = 2
A(n) = A(n-1) - 5
D. A(n) = A(n - 1) - 2
Well, this recursive formula sure knows how to subtract some numbers!
To find the recursive formula for an arithmetic sequence, we need to identify the relationship between consecutive terms.
In the given explicit formula A(n) = 2 − 5(n − 1), we can see that A(n) is expressed in terms of the previous term A(n−1) as well as a constant term.
The constant term in the explicit formula is 2, and the common difference of the arithmetic sequence is -5 (since A(n) is decreasing by 5 each time).
Therefore, the recursive formula for the arithmetic sequence is:
A(n) = A(n - 1) - 5
So the correct answer is option C: A(n) = A(n − 1) − 5.
To find the recursive formula for an arithmetic sequence, we need to express each term in terms of the previous term.
The explicit formula for this arithmetic sequence is given as A(n) = 2 − 5(n − 1).
To determine the recursive formula, we should compare two consecutive terms in the sequence. Let's compare the terms A(n) and A(n - 1):
A(n) = 2 - 5(n - 1)
A(n - 1) = 2 - 5((n - 1) - 1)
Now we can simplify both equations:
A(n) = 2 - 5n + 5
A(n - 1) = 2 - 5n + 5
We can observe that both equations are the same. Therefore, the recursive formula for this arithmetic sequence is:
A(n) = A(n - 1) + 0
In this case, the coefficient of (n - 1) is 0, which means there is no change in the sequence as we move from one term to the next. Hence, the answer is:
D. A(n) = A(n - 1) - 2