Find The 8th Term Of The Geometric Sequence

Finding the 8th Term of a Geometric Sequence: A Comprehensive Guide

Finding the nth term of a geometric sequence is a fundamental concept in algebra. This guide provides a thorough explanation of how to find the 8th term, or any term for that matter, of a geometric sequence, covering the underlying principles, step-by-step procedures, and addressing common questions. Understanding geometric sequences is crucial for various applications in mathematics, science, and finance. This article will equip you with the knowledge and skills to confidently tackle such problems.

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. This common ratio is denoted by 'r'.

For example, the sequence 2, 6, 18, 54... is a geometric sequence. Here, the first term (a₁) is 2, and the common ratio (r) is 3 (since 6 = 2 x 3, 18 = 6 x 3, and so on). The terms are related by the formula: aₙ = a₁ * r⁽ⁿ⁻¹⁾, where aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the term number.

The Formula for the nth Term of a Geometric Sequence

The core formula for calculating any term in a geometric sequence is:

aₙ = a₁ * r⁽ⁿ⁻¹⁾

Where:

• aₙ represents the nth term of the sequence (the term you want to find).
• a₁ represents the first term of the sequence.
• r represents the common ratio.
• n represents the term number (the position of the term in the sequence).

This formula provides a direct and efficient method to calculate any term in the sequence, provided you know the first term and the common ratio.

Steps to Find the 8th Term

Let's break down the process of finding the 8th term (a₈) of a geometric sequence into clear, actionable steps:

Step 1: Identify the first term (a₁) and the common ratio (r).

This is the crucial first step. You must be given, or be able to determine, both the first term and the common ratio from the sequence. For instance, consider the geometric sequence: 3, 6, 12, 24...

• a₁ = 3 (the first term)
• r = 2 (the common ratio; each term is multiplied by 2 to get the next term)

Step 2: Substitute the values into the formula.

Once you have a₁ and r, simply plug these values along with n = 8 (since we want the 8th term) into the formula:

aₙ = a₁ * r⁽ⁿ⁻¹⁾

a₈ = 3 * 2⁽⁸⁻¹⁾

Step 3: Simplify the expression.

Now, carefully simplify the expression to calculate the 8th term.

a₈ = 3 * 2⁷ a₈ = 3 * 128 a₈ = 384

Therefore, the 8th term of the geometric sequence 3, 6, 12, 24... is 384.

Examples with Different Scenarios

Let's explore a few more examples to solidify your understanding and demonstrate how to handle various situations:

Example 1: Finding the 8th term when the common ratio is a fraction.

Consider the geometric sequence: 100, 50, 25, 12.5...

• a₁ = 100
• r = 1/2 (each term is multiplied by 1/2 to get the next term)
• n = 8

Using the formula:

a₈ = 100 * (1/2)⁽⁸⁻¹⁾ a₈ = 100 * (1/2)⁷ a₈ = 100 * (1/128) a₈ = 100/128 = 25/32

Example 2: Finding the 8th term when the common ratio is negative.

Consider the geometric sequence: 1, -3, 9, -27...

• a₁ = 1
• r = -3 (each term is multiplied by -3 to get the next term)
• n = 8

Using the formula:

a₈ = 1 * (-3)⁽⁸⁻¹⁾ a₈ = 1 * (-3)⁷ a₈ = 1 * (-2187) a₈ = -2187

Example 3: Finding the 8th term when only some terms are given.

Let's say you are only given the 3rd term (a₃ = 12) and the 5th term (a₅ = 48) of a geometric sequence. How do you find the 8th term?

First, find the common ratio (r):

a₅ = a₃ * r² (since a₅ is two terms away from a₃)

48 = 12 * r² r² = 4 r = ±2 (There are two possible common ratios)

Now, we need to find a₁. We can use a₃ = a₁ * r²:

12 = a₁ * (±2)² = 4a₁ a₁ = 3 (for r=2) or a₁ = 3 (for r=-2)

Case 1 (r=2): a₈ = 3 * 2⁷ = 384

Case 2 (r=-2): a₈ = 3 * (-2)⁷ = -384

In this scenario, there are two possible geometric sequences, leading to two different 8th terms (384 and -384).

Geometric Sequences in Real-World Applications

Geometric sequences are not just abstract mathematical concepts; they have numerous practical applications:

• Compound Interest: The growth of money invested with compound interest follows a geometric sequence. Each period, the interest earned is added to the principal, and the next period's interest is calculated on the larger amount.

• Population Growth/Decay: Under certain conditions, population growth or radioactive decay can be modeled using geometric sequences.

• Spread of Diseases: In simplified models, the spread of a disease can be approximated using a geometric sequence, although real-world scenarios are far more complex.

• Computer Algorithms: Certain algorithms and data structures utilize the principles of geometric sequences for efficiency.

Frequently Asked Questions (FAQ)

Q: What if the common ratio (r) is 0?

A: If the common ratio is 0, the sequence is not a geometric sequence. A geometric sequence requires a non-zero common ratio. All terms after the first would be 0.

Q: What if the common ratio (r) is 1?

A: If the common ratio is 1, the sequence is a constant sequence (all terms are equal to the first term). The formula will still work, but the result will be the same as the first term for all values of 'n'.

Q: Can I use this formula to find earlier terms in the sequence?

A: Absolutely! The formula works for any positive integer value of 'n'. You can use it to find any term in the sequence, whether it's before, after, or even the first term (though finding the first term is usually given or can be derived from other information).

Q: What if I don't know the first term or the common ratio directly?

A: In some cases, you might be given other information, such as two terms in the sequence, from which you can deduce the common ratio and then the first term. The examples above show how to handle such scenarios.

Conclusion

Finding the 8th term (or any term) of a geometric sequence is a straightforward process once you understand the fundamental formula: aₙ = a₁ * r⁽ⁿ⁻¹⁾. By systematically identifying the first term and common ratio, and then substituting these values into the formula, you can accurately calculate the desired term. Remember to pay close attention to the sign of the common ratio, as it can affect the signs of the terms in the sequence. Mastering this concept opens up a wider understanding of sequences and their various applications in mathematics and beyond. Through practice and the application of the steps detailed above, you will gain confidence and proficiency in solving problems related to geometric sequences. Remember to always double-check your calculations to ensure accuracy.