Question

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:



















Geometric Sequences:

















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 , identify the y-intercept and explain its meaning.












Use the explicit formula to identify:

What type of sequence is it? Arithmetic or Geometric?





What is the common difference or common ratio?









What is the first term in the sequence?







What is ?
answer the last one only

Answer 1

Let's break down the questions step by step.

### Arithmetic Sequences

**Explicit Formula**:
An arithmetic sequence can be expressed explicitly as:
\[ a_n = a_1 + (n - 1)d \]
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

**Recursive Formula**:
The recursive formula for an arithmetic sequence is:
\[ a_n = a_{n-1} + d \]
where \( a_n \) is the nth term, \( a_{n-1} \) is the previous term, and \( d \) is the common difference.

**Description and Graph**:
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The graph of an arithmetic sequence is a straight line, as the terms increase (or decrease) by the same amount.

### Geometric Sequences

**Explicit Formula**:
A geometric sequence can be expressed explicitly as:
\[ a_n = a_1 \cdot r^{(n - 1)} \]
where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( r \) is the common ratio.

**Recursive Formula**:
The recursive formula for a geometric sequence is:
\[ a_n = a_{n-1} \cdot r \]
where \( a_n \) is the nth term, \( a_{n-1} \) is the previous term, and \( r \) is the common ratio.

**Description and Graph**:
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant (the common ratio). The graph of a geometric sequence is typically an exponential curve.

### T-Shirt Launch Problem

When the T-shirts are launched, the function that models the height \( h(t) \) at time \( t \) (in seconds) could be given by:
\[ h(t) = -16t^2 + 30t + 2 \]
This equation represents a parabola opening downwards due to the \( -16t^2 \) term, with an initial upward velocity and a starting height of 2 feet.

**Y-intercept**:
To find the y-intercept, we set \( t = 0 \):
\[ h(0) = -16(0)^2 + 30(0) + 2 = 2 \]
- **Meaning**: The y-intercept of 2 feet indicates the initial height of the golf cart from which T-shirts are launched.

### Sequence Analysis Request

If you are provided an explicit formula, you may analyze the sequence as follows:

1. **What type of sequence is it?**
If the formula is of the form \( a_n = a_1 + (n - 1)d \), it is an **Arithmetic** sequence.
If the formula is of the form \( a_n = a_1 \cdot r^{(n - 1)} \), it is a **Geometric** sequence.

2. **What is the common difference or common ratio?**
- For arithmetic, it's \( d \).
- For geometric, it's \( r \).

3. **What is the first term in the sequence?**
- It is \( a_1 \), the constant from the formula.

4. **What is ?**
This would refer to the specific element of the sequence which can be defined using the explicit formula above depending on \( n \).

If you provide a specific explicit formula, I can help identify the sequence type, the common ratio or difference, the first term, and give an example with computations.