which sequences are geometric select three options

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19-02-2025

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Which Sequences Are Geometric? Select Three Options

Geometric sequences are characterized by a constant ratio between consecutive terms. This means you multiply by the same number to get from one term to the next. Let's explore how to identify them and solve this multiple-choice question.

Understanding Geometric Sequences

A geometric sequence is a list of numbers where each term (after the first) is found by multiplying the previous term by a constant. This constant is called the common ratio.

For example, in the sequence 2, 6, 18, 54..., the common ratio is 3 (because 2 x 3 = 6, 6 x 3 = 18, and so on).

How to Identify a Geometric Sequence:

1. Calculate the ratio between consecutive terms: Divide each term by the term before it.
2. Check for consistency: If the ratio remains the same throughout the sequence, it's a geometric sequence.

Let's look at some examples:

Example 1: 2, 4, 8, 16, 32...

• 4/2 = 2
• 8/4 = 2
• 16/8 = 2
• 32/16 = 2

The ratio is consistently 2. This is a geometric sequence.

Example 2: 1, 3, 6, 10, 15...

• 3/1 = 3
• 6/3 = 2
• 10/6 = 1.66...
• 15/10 = 1.5

The ratio is not consistent. This is not a geometric sequence. This is actually an arithmetic sequence.

Example 3: 100, 50, 25, 12.5, 6.25...

• 50/100 = 0.5
• 25/50 = 0.5
• 12.5/25 = 0.5
• 6.25/12.5 = 0.5

The ratio is consistently 0.5. This is a geometric sequence.

Identifying Geometric Sequences: A Multiple Choice Approach

Now, let's say you're presented with a multiple-choice question like this:

Which of the following sequences are geometric? Select three options.

(Assume the options are provided here. To make this a complete problem, you would need to replace these with actual numerical sequences.)

• Option A: [Sequence 1]
• Option B: [Sequence 2]
• Option C: [Sequence 3]
• Option D: [Sequence 4]
• Option E: [Sequence 5]

To solve this, you would apply the steps outlined above to each sequence (A through E). Calculate the ratio between consecutive terms and check for consistency. Only the sequences with a constant ratio are geometric sequences. Select the three options that fit this criteria.

Remember, the key to identifying a geometric sequence is that consistent common ratio between consecutive terms. Don't be thrown off by negative common ratios or ratios that are fractions – they are still geometric sequences as long as they are consistent.