which sequences are geometric select three options

2 min read 10-03-2025

Which Sequences Are Geometric? Select Three Options

Understanding geometric sequences is crucial in mathematics. A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant value, called the common ratio. This article will help you identify geometric sequences from a selection. We'll explore several examples, highlighting the key characteristics that define a geometric sequence and helping you confidently select the correct options.

What is a Geometric Sequence?

A geometric sequence is characterized by a constant ratio between consecutive terms. This ratio, often denoted as 'r', remains the same throughout the entire sequence. To determine if a sequence is geometric, simply divide any term by its preceding term. If the result is consistent, you have a geometric sequence.

Formula: The nth term of a geometric sequence can be calculated using the formula: a_n = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number.

Identifying Geometric Sequences: Examples

Let's look at several sequences to illustrate how to identify a geometric sequence. We'll focus on three key aspects:

Consistent Ratio: The most important characteristic. Is there a consistent value obtained when dividing consecutive terms?
First Term: The sequence starts with a specific value, often denoted as a₁.
Common Ratio: The constant multiplier between consecutive terms.

Example 1: 2, 6, 18, 54, ...

Divide 6 by 2: 3
Divide 18 by 6: 3
Divide 54 by 18: 3

The common ratio is 3. This is a geometric sequence.

Example 2: 1, 4, 9, 16, ...

Divide 4 by 1: 4
Divide 9 by 4: 2.25
Divide 16 by 9: 1.78

The ratio is not consistent. This is not a geometric sequence. This is actually a quadratic sequence.

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

Divide 50 by 100: 0.5
Divide 25 by 50: 0.5
Divide 12.5 by 25: 0.5

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

Example 4: 5, 10, 20, 40, 80 ...

Divide 10 by 5 = 2
Divide 20 by 10 = 2
Divide 40 by 20 = 2
Divide 80 by 40 = 2

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

Example 5: -2, 4, -8, 16, ...

Divide 4 by -2 = -2
Divide -8 by 4 = -2
Divide 16 by -8 = -2

The common ratio is -2. This is a geometric sequence. Note that the common ratio can be negative.

How to Solve "Which Sequences Are Geometric? Select Three Options" Problems

To effectively answer multiple-choice questions asking you to identify geometric sequences, follow these steps:

Calculate the Ratio: For each sequence provided, divide each term by the previous term.
Check for Consistency: Is the ratio the same for all pairs of consecutive terms?
Select the Consistent Ones: Only the sequences with a consistent ratio are geometric sequences.

By understanding the definition and characteristics of geometric sequences and practicing with examples, you'll be able to confidently identify them and select the correct options in any multiple-choice question. Remember to always look for that consistent common ratio!

which sequences are geometric select three options