How To Find Nth Term Of A Geometric Sequence

How to Find the nth Term of a Geometric Sequence: A Comprehensive Guide

Geometric sequences are a fundamental concept in mathematics with applications spanning various fields, from finance and economics to computer science and engineering. Understanding how to find the nth term of a geometric sequence is crucial for tackling more complex mathematical problems and for interpreting real-world phenomena that exhibit geometric growth or decay. This comprehensive guide will delve into the intricacies of geometric sequences, providing you with a robust understanding of how to calculate the nth term efficiently and accurately.

Understanding Geometric Sequences

A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, often denoted by 'r'. This common ratio is the key characteristic that defines a geometric sequence.

Example:

Consider the sequence: 2, 6, 18, 54, 162...

Here, the first term (a₁) is 2. To get the second term, we multiply the first term by 3 (2 x 3 = 6). Similarly, we multiply the second term by 3 to get the third term (6 x 3 = 18), and so on. The common ratio (r) in this sequence is 3.

The Formula for the nth Term

The nth term of a geometric sequence is calculated using a specific formula. This formula elegantly encapsulates the pattern inherent in geometric progressions, allowing for the direct calculation of any term without needing to calculate all preceding terms. The formula is:

a_n = a₁ * r^(n-1)

Where:

• a_n represents the nth term of the sequence.
• a₁ represents the first term of the sequence.
• r represents the common ratio.
• n represents the position of the term in the sequence (e.g., n=1 for the first term, n=2 for the second term, etc.).

This formula is the cornerstone of solving problems related to geometric sequences. Let's break it down:

• a₁ * r: This part represents the calculation of the second term (a₂).
• r^(n-1): This part calculates how many times the common ratio is multiplied to reach the nth term. Notice that for the first term (n=1), this exponent becomes 0 (1-1=0), and any number raised to the power of 0 equals 1. This correctly yields a₁ as the first term.

Applying the Formula: Step-by-Step Examples

Let's illustrate the formula with several examples, ranging from straightforward to more challenging scenarios.

Example 1: Simple Calculation

Find the 5th term of the geometric sequence: 3, 6, 12, 24...

1. Identify a₁ and r: a₁ = 3, r = 2 (each term is double the previous one).
2. Determine n: We want the 5th term, so n = 5.
3. Apply the formula: a₅ = 3 * 2^(5-1) = 3 * 2⁴ = 3 * 16 = 48

Therefore, the 5th term of the sequence is 48.

Example 2: Negative Common Ratio

Find the 6th term of the geometric sequence: 10, -20, 40, -80...

1. Identify a₁ and r: a₁ = 10, r = -2 (each term is multiplied by -2).
2. Determine n: n = 6.
3. Apply the formula: a₆ = 10 * (-2)^(6-1) = 10 * (-2)⁵ = 10 * (-32) = -320

The 6th term is -320. Note how the negative common ratio results in alternating positive and negative terms.

Example 3: Fractional Common Ratio

Find the 4th term of the geometric sequence: 27, 9, 3, 1...

1. Identify a₁ and r: a₁ = 27, r = 1/3 (each term is one-third of the previous one).
2. Determine n: n = 4.
3. Apply the formula: a₄ = 27 * (1/3)^(4-1) = 27 * (1/3)³ = 27 * (1/27) = 1

The 4th term is 1.

Example 4: Finding the Common Ratio

Find the 5th term of a geometric sequence where the first term is 5 and the third term is 20.

1. Find r: We know that a₃ = a₁ * r^(3-1) = a₁ * r². Substituting the known values, we get 20 = 5 * r². Solving for r, we find r² = 4, so r = ±2. There are two possibilities for the common ratio.

2. Calculate a₅ for each value of r:

   • If r = 2: a₅ = 5 * 2^(5-1) = 5 * 16 = 80
   • If r = -2: a₅ = 5 * (-2)^(5-1) = 5 * 16 = 80

In this case, both possibilities for 'r' lead to the same 5th term, even though the intermediate terms differ.

Example 5: Real-world application: Compound Interest

Suppose you invest $1000 at an annual interest rate of 5%, compounded annually. What will be the balance after 10 years?

This is a geometric sequence where:

• a₁ = 1000 (initial investment)
• r = 1.05 (1 + interest rate as a decimal)
• n = 11 (10 years + the initial year)

a₁₁ = 1000 * 1.05^(11-1) = 1000 * 1.05¹⁰ ≈ 1628.89

The balance after 10 years will be approximately $1628.89. This demonstrates the power of compound interest, a classic example of geometric growth.

Handling More Complex Scenarios

While the basic formula is straightforward, some problems might require additional steps or a deeper understanding of the concepts involved.

Finding the Common Ratio when only some terms are known:

If you are given non-consecutive terms, you can still find the common ratio. For example, if you know the mth term (a_m) and the nth term (a_n), where m < n, you can use the following formula to find r:

r^(n-m) = a_n / a_m

Identifying Geometric Sequences:

Not all sequences appear geometric at first glance. To determine if a sequence is geometric, calculate the ratio between consecutive terms. If this ratio is consistent throughout the sequence, then it's a geometric sequence.

Solving for other variables:

The formula for the nth term can be manipulated to solve for other variables, such as a₁ or r, if sufficient information is provided. This involves basic algebraic manipulation.

Conclusion

Mastering the calculation of the nth term of a geometric sequence opens doors to a deeper understanding of mathematical patterns and their real-world applications. The formula, a_n = a₁ * r^(n-1), provides a powerful tool for solving various problems involving geometric growth and decay. By working through diverse examples and understanding the underlying concepts, you'll gain confidence in applying this essential mathematical skill in various contexts. Remember to practice regularly and explore different scenarios to solidify your understanding. This comprehensive guide serves as a strong foundation for further exploration into the fascinating world of geometric sequences and their numerous applications.