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:

Question

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.