Question

Compare the terms of a geometric sequence when \(r>1\) to when \(0

Short answer

The terms in a geometric sequence increase when the common ratio \(r > 1\) and decrease when \(0 < r < 1\)

Step-by-step solution

Defining a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence can be written as: \(a_{n} = a_{1} \times r^{(n-1)}\), where \(a_{1}\) is the first term, \(a_{n}\) is the n-th term, \(r\) is the common ratio, and \(n\) is the term number.

When \(r > 1\)

When the common ratio \(r > 1\), each term of the sequence is greater than the previous term. There is a common increase in the terms, resulting in the sequence being increasing. In other words, as the term number \(n\) increases, so does the value of that term in the sequence.

When \(0 < r < 1\)

When \(0 < r < 1\), each term of the sequence is less than the previous one. There is a common decrease in the terms, resulting in the sequence being decreasing. That is, as the term number \(n\) increases, the value of that term in the sequence decreases.

Key concepts

Understanding the Common Ratio

In geometric sequences, the common ratio is a crucial factor. It determines how each term in the sequence is related to the one before it. If you take any term in the sequence, say the second term, and divide it by the first term, you'll get the common ratio. Similarly, take the third term and divide it by the second, and you'll still get the same number. This consistently shared value is what we call the common ratio, denoted as \( r \).

• If \( r > 1 \), it indicates that we are multiplying each term by a number greater than 1 to get the next term, which leads to increasing terms.
• If \( 0 < r < 1 \), it suggests multiplication by a fraction, producing smaller subsequent terms, hence a decreasing sequence.

Understanding the common ratio is key to predicting and working with geometric sequences. It tells us not only the direction of the sequence (increasing or decreasing) but also impacts the rate at which the terms change.

Exploring Increasing Sequences

An increasing sequence occurs in a geometric sequence when the common ratio \( r \) is greater than 1. In this scenario, each term in the sequence is larger than the one before. Imagine you start with a number, then multiply it by a value greater than 1 — the result is a bigger number, right? This consistent growth defines an increasing sequence.

• For example, consider a sequence with a first term of 2 and a common ratio of 3. The sequence would be 2, 6, 18, 54, and so on. Each term is growing because we are multiplying by 3, a number greater than 1.
• It's like climbing stairs: each step you take is higher than the last. In mathematical terms, if \( r \rightarrow 1+ \), then \( a_{n} \rightarrow \infty \) as \( n \) increases.

This concept of increasing sequences explains why and how each term becomes larger than the last when \( r > 1 \), providing a clear framework for growth dynamics in sequences.

Demystifying Decreasing Sequences

A decreasing sequence is observed in geometric sequences when the common ratio falls between 0 and 1. When you multiply by a number less than 1, each term that follows is smaller. This shrinking effect means as you progress through the sequence, you're actually getting smaller values.

• Consider a sequence starting at 8 with a common ratio of 0.5. The sequence would look like 8, 4, 2, 1, and so forth. Notice how each term is halved from the previous one.
• Think of it as folding a piece of paper again and again; with each fold, it becomes smaller. Mathematically, if \( r \rightarrow 0+ \), then \( a_{n} \rightarrow 0 \) as \( n \) grows.

Understanding decreasing sequences illuminates how each term diminishes in value when \( 0 < r < 1 \), and highlights the nature of contraction in sequences.