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Question

You begin doing 20 sit-ups every day in November. You do 32 sit-ups every day in December, 44 sit-ups every day in January, and so on.

Write the sequence represented by the story.
Is the sequence arithmetic or geometric?
Does it have a common difference or common ratio?
What is the common difference or common ratio?
What are the next three terms?

Answer 1

Let's analyze the story step by step.

Sequence Representation:

The sequence starts with:

• November: 20 sit-ups
• December: 32 sit-ups
• January: 44 sit-ups

Following this pattern, we can see that the number of sit-ups increases each month.

The terms of the sequence are:

• First term \( a_1 = 20 \)
• Second term \( a_2 = 32 \)
• Third term \( a_3 = 44 \)

To write out the terms more clearly, we note that:

• \( a_4 = 56 \) (following the established pattern)
• \( a_5 = 68 \)
• \( a_6 = 80 \)

Sequence Formula:

Now we can represent the sequence as: \[ 20, 32, 44, 56, 68, 80, \ldots \]

Type of Sequence:

To determine if this sequence is arithmetic or geometric, we'll look for a consistent additive or multiplicative pattern.

1. Arithmetic Sequence: If the difference between consecutive terms is constant.
2. Geometric Sequence: If the ratio between consecutive terms is constant.

Analysis:

• The difference from 20 to 32 is \( 32 - 20 = 12 \).
• The difference from 32 to 44 is \( 44 - 32 = 12 \).
• The difference from 44 to 56 is \( 56 - 44 = 12 \).

Since the differences between consecutive terms are constant, this sequence is arithmetic.

Common Difference or Ratio:

• Common Difference: The difference between consecutive terms.
• Common Ratio: The ratio of consecutive terms (used only in geometric sequences).

In this case:

• Common Difference: \( d = 12 \)

Finding the Next Three Terms:

Now we can find the next three terms in the sequence:

• From 80, the next term will be \( 80 + 12 = 92 \).
• The term after that will be \( 92 + 12 = 104 \).
• The third subsequent term will be \( 104 + 12 = 116 \).

Summary of Responses:

1. Sequence: \( 20, 32, 44, 56, 68, 80, \ldots \)
2. Type: Arithmetic
3. Common Difference: \( 12 \)
4. Next Three Terms: \( 92, 104, 116 \)