Question

The first term in an arithmetic sequence is 5. The third term in the sequence is -1. The tenth term in the sequence is -22. Xavier said that the recursive formula for this sequence could be described as, 'to find the next term of the sequence, add -3 to the previous term.' Do you agree or disagree that this statement is equivalent to the explicit formula from Part A? Explain.

Answer 1

The recursive formula given by Xavier is not equivalent to the explicit formula for this arithmetic sequence.

Step-by-step explanation:

The student is asking if a recursive formula given by Xavier is equivalent to the explicit formula given earlier. The recursive formula states that to find the next term of the sequence, you add -3 to the previous term.

The explicit formula for an arithmetic sequence is given by an = a1 + (n-1)d where an is the nth term, a1 is the first term, and d is the common difference.

To determine if Xavier's recursive formula is equivalent to the explicit formula, we can check if the recursive formula produces the same sequence as the given sequence.

Let's test this by finding the terms of the sequence using both the explicit and recursive formulas:

1. a1 = 5 (given)
2. d = -1 - 5 = -6 (difference between the first and third term)
3. a3 = 5 + 2d = 5 + 2(-6) = -7 (using the recursive formula)
4. a4 = -7 - 3 = -10 (using the recursive formula)
5. a5 = -10 - 3 = -13 (using the recursive formula)
6. Continue finding more terms using the recursive formula until you reach the 10th term and compare it to the given 10th term in the sequence.

Based on the values obtained, it can be concluded that Xavier's recursive formula does not produce the same sequence as the given arithmetic sequence. Therefore, Xavier's statement is not equivalent to the explicit formula.