Geometric Sequence Problem: Find The 8th Term

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Let's break down this geometric sequence problem step by step. We're given two crucial pieces of information: the relationship between the first and third terms, and the sum of the second and third terms. Our mission is to find the first term, the common ratio, and ultimately, the eighth term of this sequence. Let's dive in!

Understanding Geometric Sequences

Before we get our hands dirty with calculations, let's refresh our understanding of geometric sequences. A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant value, called the common ratio (often denoted as 'r').

So, if the first term is 'a', the sequence looks like this:

a, ar, ar^2, ar^3, ... and so on.

The nth term of a geometric sequence is given by the formula:

a_n = a * r^(n-1)

Where:

  • a_n is the nth term
  • a is the first term
  • r is the common ratio
  • n is the term number

Now that we've got the basics covered, let's tackle the problem at hand.

Setting Up the Equations

From the problem statement, we can extract two equations:

  1. "The third term is 6 more than the first term":

    ar^2 = a + 6

  2. "The sum of the second and third terms is 9":

    ar + ar^2 = 9

These two equations form a system of equations that we can solve to find the values of 'a' (the first term) and 'r' (the common ratio).

Solving for 'a' and 'r'

Let's rearrange the first equation to isolate 'a':

ar^2 - a = 6 a(r^2 - 1) = 6 a = 6 / (r^2 - 1)

Now, substitute this expression for 'a' into the second equation:

(6 / (r^2 - 1)) * r + (6 / (r^2 - 1)) * r^2 = 9

Multiply both sides by (r^2 - 1) to get rid of the fractions:

6r + 6r^2 = 9(r^2 - 1) 6r + 6r^2 = 9r^2 - 9

Rearrange the equation into a quadratic equation:

3r^2 - 6r - 9 = 0

Divide the entire equation by 3 to simplify:

r^2 - 2r - 3 = 0

Factor the quadratic equation:

(r - 3)(r + 1) = 0

This gives us two possible solutions for 'r':

r = 3 or r = -1

Now, let's find the corresponding values of 'a' for each value of 'r':

  • If r = 3:

    a = 6 / (3^2 - 1) = 6 / 8 = 3/4

  • If r = -1:

    a = 6 / ((-1)^2 - 1) = 6 / 0 (This is undefined, so r cannot be -1)

Therefore, the only valid solution is:

a = 3/4 and r = 3

So, the first term of the geometric sequence is 3/4, and the common ratio is 3. Woohoo! We solved the first part.

Finding the 8th Term

Now that we know 'a' and 'r', we can find the 8th term (a_8) using the formula:

a_n = a * r^(n-1)

In this case, n = 8, a = 3/4, and r = 3.

a_8 = (3/4) * 3^(8-1) a_8 = (3/4) * 3^7 a_8 = (3/4) * 2187 a_8 = 6561 / 4 a_8 = 1640.25

Therefore, the 8th term of the geometric sequence is 1640.25. And that's it! We've successfully found both the first term and the 8th term of the given geometric sequence.

Alternative Approach Considerations

While we've successfully solved the problem, it's worth noting that in some geometric sequence problems, you might encounter scenarios with complex or imaginary common ratios. Always be mindful of the context and whether such solutions are plausible.

For instance, if the problem specified that the sequence consists of only real numbers, you'd need to discard any solutions involving imaginary numbers. Additionally, if the problem had further constraints (e.g., the terms must be integers), you'd need to check if your solutions satisfy those conditions.

Understanding the nature and constraints of the sequence is crucial for arriving at a valid and meaningful answer. This is a math class and your teacher is looking for a proper understanding, guys!

Key Takeaways

Let's recap the key steps we took to solve this geometric sequence problem:

  1. Understanding the Basics: We started by revisiting the definition and properties of geometric sequences, including the formula for the nth term.
  2. Setting Up Equations: We translated the given information into a system of equations involving the first term ('a') and the common ratio ('r').
  3. Solving for 'a' and 'r': We used algebraic manipulation to solve the system of equations, finding the values of 'a' and 'r'. Remember to check for extraneous solutions.
  4. Finding the 8th Term: Once we had 'a' and 'r', we plugged them into the formula for the nth term to calculate the 8th term.
  5. Considering Alternative Approaches: We discussed the importance of considering the context and constraints of the problem, especially when dealing with complex or imaginary common ratios.

By following these steps and keeping the core concepts in mind, you'll be well-equipped to tackle a wide range of geometric sequence problems!

Practice Problems

Want to test your understanding? Here are a few practice problems similar to the one we just solved:

  1. In a geometric sequence, the second term is 4, and the fourth term is 16. Find the first term and the common ratio.
  2. The sum of the first three terms of a geometric sequence is 21, and the first term is 3. Find the common ratio and the fourth term.
  3. The fifth term of a geometric sequence is 48, and the common ratio is 2. Find the first term and the sum of the first five terms.

Work through these problems, and don't hesitate to review the steps and concepts we covered in this article. With practice, you'll become a geometric sequence master!

Conclusion

Geometric sequences are a fundamental topic in mathematics with numerous applications in various fields. By mastering the concepts and techniques discussed in this article, you'll gain a solid foundation for tackling more advanced mathematical problems. So, keep practicing, keep exploring, and keep having fun with math!

Remember guys, the key to success in math is practice. The more you practice, the better you'll become. So, don't be afraid to tackle challenging problems, and always seek help when you need it. Happy problem-solving!