Finding The 7th Term Of A Geometric Sequence

Let's dive into the fascinating world of geometric sequences! In this article, we'll explore how to find a specific term within a geometric sequence, using the example sequence: $-3x^2, 3x^7, -3x^{12}, ...$ We'll break down the steps in a clear, easy-to-understand manner, so you can confidently tackle similar problems.

Understanding Geometric Sequences

To begin, it's crucial to grasp the fundamental concept of geometric sequences. A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio. This consistent multiplication factor is what sets geometric sequences apart from other types of sequences. Recognizing this pattern is key to solving problems related to these sequences.

Identifying the Common Ratio

The common ratio is the heart of a geometric sequence. To find it, you simply divide any term by its preceding term. This ratio remains constant throughout the sequence. Let's apply this to our example sequence: $-3x^2, 3x^7, -3x^{12}, ...$

To find the common ratio (r), we can divide the second term ($3x^7$) by the first term ($-3x^2$):

$r = \frac{3x^7}{-3x^2} = -x^5$

We can verify this by dividing the third term ($-3x^{12}$) by the second term ($3x^7$):

$r = \frac{-3x^{12}}{3x^7} = -x^5$

As you can see, the common ratio (r) is indeed $-x^5$. This consistent ratio is what defines this sequence as geometric and is essential for finding subsequent terms.

The General Formula for Geometric Sequences

Now that we understand the common ratio, let's introduce the general formula that governs geometric sequences. This formula allows us to calculate any term in the sequence directly, without having to compute all the preceding terms. The general formula for the nth term ($a_n$) of a geometric sequence is:

$a_n = a_1 * r^(n-1)$

Where:

• $a_n$ is the nth term we want to find.
• $a_1$ is the first term of the sequence.
• r is the common ratio.
• n is the term number we are looking for.

This formula is a powerful tool. It encapsulates the essence of geometric progression and provides a direct route to finding any term, given the first term, the common ratio, and the term number.

Finding the 7th Term

Now, let's put our knowledge into action and find the 7th term of the sequence $-3x^2, 3x^7, -3x^{12}, ...$. We have all the necessary components: the first term, the common ratio, and the term number we're interested in.

Applying the Formula

We know:

• $a_1 = -3x^2$ (the first term)
• $r = -x^5$ (the common ratio)
• $n = 7$ (we want to find the 7th term)

Plugging these values into the general formula, we get:

$a_7 = -3x^2 * (-x^5)^(7-1)$

Step-by-Step Calculation

Let's break down the calculation step-by-step:

1. Simplify the exponent: $7 - 1 = 6$

$a_7 = -3x^2 * (-x^5)^6$

2. Apply the exponent to the common ratio: $(-x^5)^6 = x^{30}$ (Remember that a negative number raised to an even power becomes positive)

$a_7 = -3x^2 * x^{30}$

3. Multiply the terms: When multiplying exponents with the same base, we add the powers: $x^2 * x^{30} = x^{32}$

$a_7 = -3x^{32}$

Therefore, the 7th term of the geometric sequence is $-3x^{32}$. By applying the formula and carefully following the steps, we have successfully determined the 7th term.

Key Takeaways

Let's recap the essential steps we've learned for finding a specific term in a geometric sequence. These key takeaways will serve as a valuable guide for solving similar problems in the future.

1. Identify the sequence as geometric: Ensure that there is a constant common ratio between consecutive terms.
2. Determine the first term ($a_1$) and the common ratio (r): These are the fundamental building blocks of the sequence.
3. Use the general formula: $a_n = a_1 * r^(n-1)$ This formula is your primary tool for finding any term.
4. Substitute the values: Carefully plug in the values for $a_1$, r, and n.
5. Simplify the expression: Follow the order of operations to calculate the desired term.

By mastering these steps, you'll be well-equipped to handle a wide range of geometric sequence problems. The general formula, in particular, is a powerful tool that can save you time and effort.

Practice Makes Perfect

The best way to solidify your understanding of geometric sequences is through practice. Try applying these steps to various geometric sequences. You can create your own sequences or find examples in textbooks or online resources. The more you practice, the more confident you'll become in your ability to solve these problems.

Example Practice Problems

Here are a couple of practice problems to get you started:

1. Find the 10th term of the sequence: 2, 6, 18, ...
2. Find the 6th term of the sequence: 5, -10, 20, ...

Work through these problems, applying the steps we've discussed. Check your answers against solutions if available. This active engagement with the material will significantly enhance your understanding and retention.

Conclusion

Finding a specific term in a geometric sequence is a straightforward process when you understand the underlying principles and the general formula. By identifying the common ratio, applying the formula, and practicing regularly, you can confidently solve these types of problems. Geometric sequences are a fascinating area of mathematics, and mastering them opens doors to more advanced concepts.

For further exploration of geometric sequences and related topics, you can visit Khan Academy's section on geometric sequences.