Problem 18 - College Algebra

Question

For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio. The first term is 2 , and the common ratio is 3 . Find the \(5^{\text {th }}\) term.

Step-by-Step Solution

Answer

The 5th term is 162.

Step 1: Identify the Geometric Sequence Formula

The formula for the nth term of a geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

Step 2: Substitute Given Values

Here, \( a_1 = 2 \) and \( r = 3 \). We need to find the 5th term, so \( n = 5 \). Substitute these values into the formula: \( a_5 = 2 \cdot 3^{(5-1)} \).

Step 3: Calculate the Power of the Common Ratio

Calculate \( 3^{4} \): \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \).

Step 4: Find the 5th Term

Substitute the calculated power back into the formula: \( a_5 = 2 \cdot 81 = 162 \).

Key Concepts

nth term

In a geometric sequence, each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula for finding the nth term of a geometric progression is given by \( a_n = a_1 \cdot r^{(n-1)} \).

• \( a_n \) represents the nth term that is being calculated.
• \( a_1 \) is the first term of the sequence.
• \( r \) refers to the common ratio.
• \( n \) indicates the position of the term within the sequence.

This formula enables you to find any term in the sequence without having to list all the terms up to that point. It is particularly useful when you're dealing with large sequences.

common ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It is crucial because it governs the growth or decay pattern of the sequence.
If the common ratio \( r \) is greater than 1, the terms increase, forming an increasing geometric progression.

• For example, if \( r = 3 \), it means every term is three times the previous one.
• If \( r = 1/2 \), each term is half of the previous, leading to a decreasing sequence.

Knowing the common ratio helps you easily extend the sequence or find any specific term using the formula for the nth term.

geometric progression

A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, constant value known as the common ratio. This sequence is defined mathematically through the general formula \( a_n = a_1 \cdot r^{(n-1)} \), where:

• The sequence starts at \( a_1 \), the first term.
• Each consecutive term is the product of the previous term and the common ratio \( r \).

Understanding geometric progressions is fundamental in various fields, such as finance for compound interest calculations, in physics for wave patterns, and other real-world applications.

term calculation

Calculating a specific term in a geometric sequence involves using the nth term formula, where you insert the values for the first term \( a_1 \), the common ratio \( r \), and the term number \( n \).
For instance, if you're tasked to find the 5th term of a sequence where \( a_1 = 2 \) and \( r = 3 \), use the formula:\[ a_5 = 2 \cdot 3^{(5-1)} \].

• First, compute \( 3^{(5-1)} = 3^4 = 81 \).
• Then, multiply by the first term: \( 2 \cdot 81 = 162 \).

This approach simplifies finding any term without sequentially calculating each preceding term, saving you time on lengthy sequences.