Using the Formula for the Limit of a Geometric Sequence

Oct 13, 2025 by wbrum.org 61 views

Introduction

In mathematics, a sequence is a list of numbers in a specific order. The limit of a sequence is the value that the sequence approaches as the number of terms increases without bound. In this article, we will determine the limit of a given geometric sequence.

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general formula for a geometric sequence is:

a_n = a_1 \cdot r^{n-1}

where $a_n$ is the nth term of the sequence, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

The Given Sequence

The given sequence is:

c_n = \left(\frac{3}{2}\right)^n

This is a geometric sequence with a common ratio of $\frac{3}{2}$ .

Determining the Limit

To determine the limit of the sequence, we need to find the value that the sequence approaches as $n$ approaches infinity. In other words, we need to find the value of $c_n$ as $n$ becomes very large.

Using the Formula for the Limit of a Geometric Sequence

The formula for the limit of a geometric sequence is:

\lim_{n \to \infty} a_n = \begin{cases} \infty & \text{if } |r| > 1 \\ 0 & \text{if } |r| < 1 \\ a_1 & \text{if } r = 1 \end{cases}

In this case, the common ratio $r$ is $\frac{3}{2}$ , which is greater than 1. Therefore, the limit of the sequence is infinity.

Conclusion

The limit of the given geometric sequence is infinity. This means that as the number of terms increases without bound, the value of the sequence approaches infinity.

Final Answer

The final answer is:

\lim_{n \to \infty} c_n = \infty

Discussion

The limit of a geometric sequence depends on the value of the common ratio. If the common ratio is greater than 1, the limit is infinity. If the common ratio is less than 1, the limit is 0. If the common ratio is 1, the limit is the first term of the sequence.

Example Problems

Find the limit of the sequence $a_n = 2 \cdot 3^n$ .
Find the limit of the sequence $b_n = \left(\frac{1}{2}\right)^n$ .
Find the limit of the sequence $c_n = 5 \cdot 2^n$ .

Solutions

The limit of the sequence $a_n = 2 \cdot 3^n$ is infinity, since the common ratio is greater than 1.
The limit of the sequence $b_n = \left(\frac{1}{2}\right)^n$ is 0, since the common ratio is less than 1.
The limit of the sequence $c_n = 5 \cdot 2^n$ is infinity, since the common ratio is greater than 1.

Practice Problems

Find the limit of the sequence $a_n = 4 \cdot 5^n$ .
Find the limit of the sequence $b_n = \left(\frac{1}{3}\right)^n$ .
Find the limit of the sequence $c_n = 2 \cdot 4^n$ .

Solutions

The limit of the sequence $a_n = 4 \cdot 5^n$ is infinity, since the common ratio is greater than 1.
The limit of the sequence $b_n = \left(\frac{1}{3}\right)^n$ is 0, since the common ratio is less than 1.
The limit of the sequence $c_n = 2 \cdot 4^n$ is infinity, since the common ratio is greater than 1.