Problem 24 - Algebra and Trigonometry

Question

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=1 ; \quad r=-\frac{1}{3} $$

Step-by-Step Solution

Answer

The fifth term is \( \frac{1}{81} \) and the nth term is \( \left( -\frac{1}{3} \right)^{n-1} \).

Step 1 - Understand the Geometric Sequence Basics

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant called the common ratio, denoted as \( r \).

Step 2 - Identify Given Values

The problem provides the first term \( a_1 = 1 \) and the common ratio \( r = -\frac{1}{3} \).

Step 3 - Find the Fifth Term

Use the formula for the nth term of a geometric sequence, which is given by \( a_n = a_1 \cdot r^{n-1} \). Substitute \( n = 5 \), \( a_1 = 1 \), and \( r = -\frac{1}{3} \) to get the 5th term: \[ a_5 = 1 \cdot \left(-\frac{1}{3}\right)^{5-1} = \left(-\frac{1}{3}\right)^4 = \left(\frac{1}{3}\right)^4 = \frac{1}{81} \].

Step 4 - Find the nth Term

To find the general nth term, use the same formula \( a_n = a_1 \cdot r^{n-1} \). Substitute the given values \( a_1 = 1 \) and \( r = -\frac{1}{3} \): \[ a_n = 1 \cdot \left(-\frac{1}{3}\right)^{n-1} = \left(-\frac{1}{3}\right)^{n-1} \].

Key Concepts

First Term

In any geometric sequence, the first term is the starting point of the sequence and is usually denoted as \(a_1\). It is the number that begins the sequence of numbers.

This term is crucial because, along with the common ratio, it helps define the entire sequence. For instance, in the given problem, the first term is 1. This means our sequence starts from 1.

Without knowing the first term, we wouldn't be able to identify the structure of the sequence, or determine any of its other terms.

Common Ratio

The common ratio in a geometric sequence is the constant factor that each term is multiplied by to get the next term. It is usually represented by the letter \( r \).

In the given problem, the common ratio is \(-\frac{1}{3}\). This means that to get from one term to the next, you multiply by \(-\frac{1}{3}\).

The common ratio can be positive or negative, and its value determines how the sequence progresses. If the ratio is negative, as in this problem, the terms will alternate in sign.

Understanding the common ratio helps in predicting the following terms and also in understanding the behavior of the sequence over time.

Nth Term Formula

The nth term formula in a geometric sequence allows you to find any term in the sequence without listing all of the previous terms.
It is given by:
\[ a_n = a_1 \times r^{(n-1)} \]
In this formula:

• \(a_n\) is the nth term.
• \(a_1\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the position of the term in the sequence.

For the given problem, the nth term can be calculated by substituting \(a_1 = 1\) and \(r = -\frac{1}{3}\):
\[ a_n = 1 \times \bigg(-\frac{1}{3}\bigg)^{(n-1)} \]
This means substituting any value for \(n\) will give you the corresponding term in the sequence.

Sequence

A sequence is an ordered list of numbers, with each number called a term. The order of the numbers is essential in defining the sequence.

In a geometric sequence, the terms are generated by starting with a given number (the first term) and multiplying it repeatedly by a fixed, non-zero number called the common ratio.

The given problem defines a geometric sequence with a first term of 1 and a common ratio of \(-\frac{1}{3}\).
This tells us that each term in the sequence is derived by multiplying the previous term by \(-\frac{1}{3}\). Understanding this structure helps predict any term in the sequence based on its position and the rules provided.