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Use the explicit formula to identify:

What type of sequence is it? Arithmetic or Geometric?

11 answers

To identify whether a sequence is arithmetic or geometric, we need to look at the explicit formula provided for the sequence. However, since you haven't provided a specific formula, I can explain the characteristics of each type:

  1. Arithmetic Sequence:

    • An arithmetic sequence has a constant difference between consecutive terms. The explicit formula for an arithmetic sequence can be written as: \[ a_n = a_1 + (n-1)d \] where \( a_n \) is the \(n\)-th term, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

  2. Geometric Sequence:

    • A geometric sequence has a constant ratio between consecutive terms. The explicit formula for a geometric sequence can be written as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( a_n \) is the \(n\)-th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

To determine the type of sequence based on a specific explicit formula, you would need to identify if the formula follows the format of an arithmetic sequence or a geometric sequence.

If you have a specific explicit formula for the sequence, please provide it, and I can help you determine whether it is arithmetic or geometric.

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To identify the type of sequence from an explicit formula:

  • Arithmetic Sequence: If the formula is of the form \( a_n = a_1 + (n-1)d \) (constant difference).

  • Geometric Sequence: If the formula is of the form \( a_n = a_1 \cdot r^{(n-1)} \) (constant ratio).

Please provide the explicit formula for a specific determination.

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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.

To identify the y-intercept of the function describing the T-shirt's motion, we typically assume a quadratic function of the form:

\[ h(t) = -16t^2 + v_0t + h_0 \]

where:

  • \( h(t) \) is the height at time \( t \),
  • \( v_0 \) is the initial velocity (30 ft/s),
  • \( h_0 \) is the initial height (2 ft off the ground).

So, the equation is:

\[ h(t) = -16t^2 + 30t + 2 \]

Y-Intercept:

The y-intercept occurs when \( t = 0 \):

\[ h(0) = -16(0)^2 + 30(0) + 2 = 2 \]

Meaning:

The y-intercept (\(2\) feet) represents the initial height of the T-shirts when launched from the golf cart.

What is the common difference or common ratio?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.

In the context of the problem you provided, we are examining the motion of T-shirts launched into the air. This scenario does not directly involve sequences with a common difference or common ratio since these terms apply to arithmetic and geometric sequences, respectively.

Understanding the Motion:

In the situation you described, we can model the height of the T-shirts as a function of time using a quadratic equation:

\[ h(t) = -16t^2 + 30t + 2 \]

Key Characteristics:

  1. Y-Intercept:

    • The y-intercept occurs when \( t = 0 \): \[ h(0) = -16(0)^2 + 30(0) + 2 = 2 \]
    • Meaning: The y-intercept of \(2\) feet signifies the initial height of the T-shirts when they are launched, indicating that the T-shirts start at a height of \(2\) feet from the ground.

  2. Common Difference/Ratio:

    • In this scenario, we do not have a common difference (as in an arithmetic sequence) or a common ratio (as in a geometric sequence). Instead, the height of the T-shirts changes in a non-linear way due to the effects of gravity over time, characterized by the quadratic formula.

Summary:

  • Common Difference/Ratio: Not applicable in this situation.
  • Y-Intercept: \(2\) feet, representing the initial launch height of the T-shirts.
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Common Difference or Ratio:

Not applicable; the motion follows a quadratic model, not a sequence.

Y-Intercept:

  • Value: 2 feet
  • Meaning: This represents the initial height of the T-shirts when launched from the golf cart.
Use the explicit formula to identify:

What type of sequence is it? Arithmetic or Geometric?

To determine the type of sequence using an explicit formula, please provide the specific formula you are referencing.

General Guidance:

  • Arithmetic Sequence: If the formula is of the form \( a_n = a_1 + (n-1)d \), where \( d \) is a constant difference.
  • Geometric Sequence: If the formula is of the form \( a_n = a_1 \cdot r^{(n-1)} \), where \( r \) is a constant ratio.

With the specific formula, I can help you identify the type of sequence!