4. Find the recursive formula for the arithmetic sequence 7, 5, 3, 1, −1...

Question

A) a1 = 7, an = an-1 + 2
B) a1 = 7, an = an-1 - 2
C) a1 = 7, an = -2an-1
D) a1 = 7, an = 2an-1

5. Which of the following correctly uses the explicit formula, An = 1/4 + (n - 1) x 1/2, to find the 9th term of the described arithmetic sequence?
A) 8 3/4
B) 4 1/4
C) 4 3/4
D) 8 1/4

Answer 1

4. To find the recursive formula for the arithmetic sequence 7, 5, 3, 1, -1..., we first observe the pattern of the sequence.

• The first term \( a_1 \) is 7.
• To get from one term to the next, we subtract 2 (e.g., \( 7 - 2 = 5 \), \( 5 - 2 = 3 \), etc.).

Therefore, the recursive formula can be written as:

• \( a_1 = 7 \)
• \( a_n = a_{n-1} - 2 \)

The correct answer is B) a1 = 7, an = an-1 - 2.

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5. To find the 9th term of the arithmetic sequence described by the explicit formula \( A_n = \frac{1}{4} + (n - 1) \times \frac{1}{2} \), we will substitute \( n = 9 \) into the formula:

\[ A_9 = \frac{1}{4} + (9 - 1) \times \frac{1}{2} \] \[ = \frac{1}{4} + 8 \times \frac{1}{2} \] \[ = \frac{1}{4} + 4 \] \[ = \frac{1}{4} + \frac{16}{4} \] \[ = \frac{1 + 16}{4} \] \[ = \frac{17}{4} \] \[ = 4 \frac{1}{4} \]

Thus, the correct answer is B) 4 1/4.