which sequences are geometric select three options

Daniel Parker

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27-02-2025

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Which Sequences Are Geometric? Selecting the Correct Options

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant. This constant is called the common ratio. Understanding this definition is key to identifying geometric sequences. Let's explore how to select the correct options when presented with a multiple-choice question asking which sequences are geometric.

Identifying Geometric Sequences: A Step-by-Step Guide

To determine if a sequence is geometric, follow these steps:

1. Calculate the ratio between consecutive terms: Divide each term by the preceding term. For example, in the sequence 2, 4, 8, 16, you would calculate 4/2 = 2, 8/4 = 2, and 16/8 = 2.

2. Check for a constant ratio: If the ratio between consecutive terms is the same throughout the sequence, then the sequence is geometric. In our example, the constant ratio is 2.

3. Consider the first term: Even if the ratios are consistent after the first term, a sequence is not geometric if there's no clear first term to start this multiplicative pattern from.

Examples: Identifying Geometric and Non-Geometric Sequences

Let's look at some examples to solidify our understanding:

Example 1: Geometric Sequence

• Sequence: 3, 6, 12, 24, 48...
• Ratio: 6/3 = 2, 12/6 = 2, 24/12 = 2, 48/24 = 2
• Conclusion: This is a geometric sequence with a common ratio of 2.

Example 2: Non-Geometric Sequence

• Sequence: 1, 3, 6, 10, 15...
• Ratios: 3/1 = 3, 6/3 = 2, 10/6 = 1.67, 15/10 = 1.5
• Conclusion: This is not a geometric sequence because the ratios between consecutive terms are not constant.

Example 3: Tricky Case – Appears Geometric but Isn't

• Sequence: 0, 0, 0, 0...
• Ratios: Division by zero is undefined.
• Conclusion: This sequence is not considered geometric. While it might seem like it has a common ratio of 0 (because 0*0 = 0), division by zero is mathematically undefined, making it not a geometric sequence.

Practice: Selecting Three Geometric Sequences

Now, let's put our knowledge to the test. Imagine you're given a multiple-choice question with several sequences. To select the three geometric sequences, you would apply the steps outlined above to each sequence. Only those sequences exhibiting a consistent, non-zero ratio between consecutive terms would be considered geometric.

Remember to carefully calculate the ratios and check for consistency across the entire sequence before making your selections. Don't be tricked by sequences that appear geometric at first glance but lack a consistent ratio throughout. Pay close attention to the first term and potential issues arising from division by zero.