. The 3rd term of an arithmetic sequence is 3/2

Question

. The common difference is 1/2
. Complete the statement.(2 points)
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The explicit formula for the sequence is
and the recursive formula for the sequence is
.
Question 2
5. The first time Mya competed in a 500 meter free style race, her time was 15 minutes flat. Each subsequent race, Mya's time was reduced by 10 seconds.

Complete the statement.

(2 points)
The context is best represented by
an arithmetic sequence
and the domain is
continuous

Answer 1

Question 1:

To find the explicit and recursive formulas for the arithmetic sequence, we start by noting that the 3rd term \( a_3 \) is given as \( \frac{3}{2} \) and the common difference \( d \) is \( \frac{1}{2} \).

1. The explicit formula for an arithmetic sequence is given by:
   \[ a_n = a_1 + (n - 1) \cdot d \]
   First, we need to find \( a_1 \) (the first term). Since we know that the 3rd term is \( a_3 = a_1 + 2d \): \[ \frac{3}{2} = a_1 + 2 \cdot \frac{1}{2} \] \[ \frac{3}{2} = a_1 + 1 \] \[ a_1 = \frac{3}{2} - 1 = \frac{1}{2} \]

   Now we can write the explicit formula:
   \[ a_n = \frac{1}{2} + (n - 1) \cdot \frac{1}{2} \]
   Simplifying this, we get:
   \[ a_n = \frac{1}{2} + \frac{n - 1}{2} = \frac{1 + n - 1}{2} = \frac{n}{2} \]

   So, the explicit formula for the sequence is \( \frac{n}{2} \).

2. The recursive formula for the arithmetic sequence can be expressed as:
   \[ a_n = a_{n-1} + d \]
   Given \( a_1 = \frac{1}{2} \) and the common difference \( d = \frac{1}{2} \):
   \[ a_n = a_{n-1} + \frac{1}{2} \text{ for } n \geq 2 \]

   Thus, the recursive formula for the sequence is:
   \( a_1 = \frac{1}{2}, , a_n = a_{n-1} + \frac{1}{2} \)

Final Answers for Question 1:
The explicit formula for the sequence is \(\frac{n}{2}\) and the recursive formula for the sequence is \(a_1 = \frac{1}{2}, , a_n = a_{n-1} + \frac{1}{2}\).

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Question 2:

Mya's race times form an arithmetic sequence where the first term (time for the first race) is 15 minutes and each subsequent term is reduced by 10 seconds.

1. The context is best represented by an arithmetic sequence since a constant amount (10 seconds) is being subtracted from the time for each race.

2. The domain of Mya's race times is best described as discrete because the number of races she competes in is a whole number (0, 1, 2, ...).

Final Answers for Question 2:
The context is best represented by an arithmetic sequence and the domain is discrete.