Question

find the recursive formula for the arithmetic sequence 7,5,3,1,-1,...
a a1 = 7, an = 2an-1
b a1 = 7, an = an-1^-2
c a1 = 7, an = -2an-1
d a1 = 7, an = an-1^+2

Answer 1

To find the recursive formula for the given arithmetic sequence \( 7, 5, 3, 1, -1, \ldots \), we need to identify the pattern in how the terms are generated from one another.

The sequence decreases by 2 each time:
- \( 7 - 2 = 5 \)
- \( 5 - 2 = 3 \)
- \( 3 - 2 = 1 \)
- \( 1 - 2 = -1 \)

This indicates that the recursive relationship involves subtracting 2 from the previous term.

Thus, the recursive formula can be expressed as:
- \( a_1 = 7 \)
- \( a_n = a_{n-1} - 2 \) for \( n \geq 2 \)

Let's compare this with the options provided:
- a. \( a_1 = 7, a_n = 2 a_{n-1} \) (incorrect, it doubles the previous term)
- b. \( a_1 = 7, a_n = a_{n-1} - 2 \) (this matches our derived formula)
- c. \( a_1 = 7, a_n = -2 a_{n-1} \) (incorrect, it multiplies by -2)
- d. \( a_1 = 7, a_n = a_{n-1} + 2 \) (incorrect, it adds 2)

The correct answer is **b**: \( a_1 = 7, a_n = a_{n-1} - 2 \).