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

Question

. The common difference is 1/2
. Complete the statement.(2 points)

Answer 1

To complete the statement about the arithmetic sequence, we can use the information provided:

• The 3rd term of the sequence is \( a_3 = \frac{3}{2} \).
• The common difference is \( d = \frac{1}{2} \).

In an arithmetic sequence, the \( n \)-th term can be calculated using the formula:

\[ a_n = a_1 + (n-1)d \]

For the 3rd term, the formula would be:

\[ a_3 = a_1 + 2d \]

Substituting the known values:

\[ \frac{3}{2} = a_1 + 2 \left( \frac{1}{2} \right) \]

This simplifies to:

\[ \frac{3}{2} = a_1 + 1 \]

Now, solving for \( a_1 \):

\[ a_1 = \frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2} \]

Now that we have \( a_1 \), we can state:

• The first term of the arithmetic sequence is \( \frac{1}{2} \).
• The second term can be found by adding the common difference to the first term:

\[ a_2 = a_1 + d = \frac{1}{2} + \frac{1}{2} = 1 \]

To summarize:

The first term of the arithmetic sequence is \( \frac{1}{2} \), the second term is \( 1 \), and the third term is \( \frac{3}{2} \). The common difference is \( \frac{1}{2} \).

Complete statement: The first term of the arithmetic sequence is \(\frac{1}{2}\), and the second term is \( 1 \).

Answer 2

The explicit formula for the sequence is Response area and the recursive formula for the sequence is

Answer 3

To express the sequence in both explicit and recursive forms, let's start by summarizing the information we have about the arithmetic sequence:

• First term \( a_1 = \frac{1}{2} \)
• Common difference \( d = \frac{1}{2} \)

Explicit Formula

The explicit formula for an arithmetic sequence is given by:

\[ a_n = a_1 + (n-1)d \]

Substituting in the values we have:

\[ a_n = \frac{1}{2} + (n-1)\left(\frac{1}{2}\right) \]

This can be simplified as follows:

\[ a_n = \frac{1}{2} + \frac{1}{2}(n-1) = \frac{1}{2} + \frac{n-1}{2} = \frac{1 + (n-1)}{2} = \frac{n}{2} \]

So, the explicit formula for the sequence is:

\[ \boxed{a_n = \frac{n}{2}} \]

Recursive Formula

The recursive formula provides the first term and a way to find subsequent terms from the previous term. It is given by:

\[ a_n = a_{n-1} + d \quad \text{for } n \geq 2 \]

Substituting the known values:

\[ a_n = a_{n-1} + \frac{1}{2} \quad \text{for } n \geq 2 \]

And we specify the first term:

\[ a_1 = \frac{1}{2} \]

Therefore, the recursive formula for the sequence is:

\[ \boxed{a_n = a_{n-1} + \frac{1}{2}, , a_1 = \frac{1}{2}} \]

Summary

1. Explicit Formula: \( a_n = \frac{n}{2} \)
2. Recursive Formula: \( a_n = a_{n-1} + \frac{1}{2}, , a_1 = \frac{1}{2} \)

Answer 4

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 _________ and the domain is _______

Answer 5

To complete the statement regarding Mya's competition times, we need to represent the context of her race times with a function and specify the domain.

1. The context: Mya's time for the 500-meter freestyle race can be represented as a function that captures her performance over time (in races). As Mya continues to compete, her time decreases by consistent intervals. Therefore, we can describe her time in the \( n \)-th race as follows:

   • The first race (when \( n = 0 \)): 15 minutes = 900 seconds.
   • Each subsequent race (for \( n \geq 1 \)): her time decreases by 10 seconds.

   Thus, the context can be best represented by the function:

   \[ T(n) = 900 - 10n \]

2. The domain: The domain of this function should include all realistic values of \( n \) (number of races). Since \( n \) must be a non-negative integer (you can't have a negative number of races), it starts at 0 and can increase indefinitely until Mya's time reaches a limit (for example, a minimum viable time). However, for practical purposes in typical scenarios like this, we can say the domain includes non-negative integers until her time becomes impractically low (e.g., we can't continue indefinitely).

   Therefore, the domain can be expressed as:

   \[ \text{Domain: } n \in {0, 1, 2, 3, \ldots} \text{ (non-negative integers)} \]

Complete Statement

The context is best represented by \( T(n) = 900 - 10n \) and the domain is \( n \in {0, 1, 2, 3, \ldots} \) (non-negative integers).