Asked by adaleta
Provide the explicit and recursive formulas for the following, as well as a description of and a drawing/example of the graph(s) for that type of sequence:
Arithmetic Sequences:
Arithmetic Sequences:
Answers
Answered by
GPT-4o mini
### Arithmetic Sequences
**Definition:**
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the common difference.
**Common Difference:**
Let \(d\) be the common difference. If \(a_1\) is the first term of the sequence, then each term can be expressed as:
- \(a_2 = a_1 + d\)
- \(a_3 = a_2 + d = a_1 + 2d\)
- and so on.
**Explicit Formula:**
The explicit formula for an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1)d
\]
where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.
**Recursive Formula:**
The recursive formula for an arithmetic sequence can be expressed as:
\[
\begin{align*}
a_1 & = \text{(first term)} \\
a_n & = a_{n-1} + d \quad \text{for } n > 1
\end{align*}
\]
**Graph of an Arithmetic Sequence:**
The graph of an arithmetic sequence is a straight line when plotted on a Cartesian coordinate system. This is because there is a constant change (the common difference) in the values of the sequence with respect to the term number.
### Example:
Let’s consider an arithmetic sequence where the first term \(a_1 = 3\) and the common difference \(d = 2\).
- The sequence would be:
\[
3, 5, 7, 9, 11, \ldots
\]
**Explicit Formula:**
\[
a_n = 3 + (n-1) \cdot 2 = 2n + 1
\]
**Recursive Formula:**
\[
\begin{align*}
a_1 & = 3 \\
a_n & = a_{n-1} + 2 \quad \text{for } n > 1
\end{align*}
\]
**Graph:**
To illustrate, we can plot the sequence on a graph:
- \(x\) (horizontal axis) represents \(n\) (term number),
- \(y\) (vertical axis) represents \(a_n\) (value of the term).
```
|
11 | * (6)
10 | *
9 | *
8 | *
7 | *
6 |
5 | *
4 |
3 | * (1)
2 |
1 |
|________________________________
1 2 3 4 5 6
```
- Each point corresponds to a term in the sequence, demonstrating the linearity and equal spacing between the points, reflective of the constant difference of \(2\).
**Definition:**
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the common difference.
**Common Difference:**
Let \(d\) be the common difference. If \(a_1\) is the first term of the sequence, then each term can be expressed as:
- \(a_2 = a_1 + d\)
- \(a_3 = a_2 + d = a_1 + 2d\)
- and so on.
**Explicit Formula:**
The explicit formula for an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1)d
\]
where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.
**Recursive Formula:**
The recursive formula for an arithmetic sequence can be expressed as:
\[
\begin{align*}
a_1 & = \text{(first term)} \\
a_n & = a_{n-1} + d \quad \text{for } n > 1
\end{align*}
\]
**Graph of an Arithmetic Sequence:**
The graph of an arithmetic sequence is a straight line when plotted on a Cartesian coordinate system. This is because there is a constant change (the common difference) in the values of the sequence with respect to the term number.
### Example:
Let’s consider an arithmetic sequence where the first term \(a_1 = 3\) and the common difference \(d = 2\).
- The sequence would be:
\[
3, 5, 7, 9, 11, \ldots
\]
**Explicit Formula:**
\[
a_n = 3 + (n-1) \cdot 2 = 2n + 1
\]
**Recursive Formula:**
\[
\begin{align*}
a_1 & = 3 \\
a_n & = a_{n-1} + 2 \quad \text{for } n > 1
\end{align*}
\]
**Graph:**
To illustrate, we can plot the sequence on a graph:
- \(x\) (horizontal axis) represents \(n\) (term number),
- \(y\) (vertical axis) represents \(a_n\) (value of the term).
```
|
11 | * (6)
10 | *
9 | *
8 | *
7 | *
6 |
5 | *
4 |
3 | * (1)
2 |
1 |
|________________________________
1 2 3 4 5 6
```
- Each point corresponds to a term in the sequence, demonstrating the linearity and equal spacing between the points, reflective of the constant difference of \(2\).
Answered by
adaleta
The cheerleaders at a football game launch T-shirts into the crowd from the back of a golf cart that is 2 feet off the ground. The T-shirts have an upward velocity of 30 feet per second. Using the function y=-16t2+30t+2, identify the y-intercept and explain its meaning.
Answered by
GPT-4o mini
The function provided to describe the motion of the T-shirts is given by:
\[
y = -16t^2 + 30t + 2
\]
where:
- \(y\) represents the height (in feet) of the T-shirt above the ground after \(t\) seconds,
- \(t\) is the time in seconds,
- The term \(-16t^2\) represents the effect of gravity (with a downward acceleration of \(32 \, \text{ft/s}^2\) affecting the upward motion),
- The term \(30t\) represents the initial upward velocity of the T-shirts,
- The constant \(2\) represents the initial height from which the T-shirts are launched (2 feet above the ground).
### Y-Intercept
The y-intercept of a function occurs where \(t = 0\). To find the y-intercept of the function, we will substitute \(t = 0\) into the equation:
\[
y = -16(0)^2 + 30(0) + 2
\]
\[
y = 0 + 0 + 2
\]
\[
y = 2
\]
Thus, the **y-intercept** is \( (0, 2) \).
### Meaning of the Y-Intercept
The y-intercept (2 feet in this case) represents the height of the T-shirts at the moment they are launched, which is the initial position of the T-shirts above the ground. Since the T-shirts are being launched from the back of a golf cart that is 2 feet off the ground, the y-intercept reflects that initial height - they are 2 feet in the air when they are first thrown.
This gives a clear context in understanding the motion of the T-shirts; they start their trajectory from that point at a height of 2 feet.
\[
y = -16t^2 + 30t + 2
\]
where:
- \(y\) represents the height (in feet) of the T-shirt above the ground after \(t\) seconds,
- \(t\) is the time in seconds,
- The term \(-16t^2\) represents the effect of gravity (with a downward acceleration of \(32 \, \text{ft/s}^2\) affecting the upward motion),
- The term \(30t\) represents the initial upward velocity of the T-shirts,
- The constant \(2\) represents the initial height from which the T-shirts are launched (2 feet above the ground).
### Y-Intercept
The y-intercept of a function occurs where \(t = 0\). To find the y-intercept of the function, we will substitute \(t = 0\) into the equation:
\[
y = -16(0)^2 + 30(0) + 2
\]
\[
y = 0 + 0 + 2
\]
\[
y = 2
\]
Thus, the **y-intercept** is \( (0, 2) \).
### Meaning of the Y-Intercept
The y-intercept (2 feet in this case) represents the height of the T-shirts at the moment they are launched, which is the initial position of the T-shirts above the ground. Since the T-shirts are being launched from the back of a golf cart that is 2 feet off the ground, the y-intercept reflects that initial height - they are 2 feet in the air when they are first thrown.
This gives a clear context in understanding the motion of the T-shirts; they start their trajectory from that point at a height of 2 feet.