Finding The 16th Term Of A Geometric Sequence A Step-by-Step Guide

Jul 31, 2025 by ADMIN 67 views

Unlocking the 16th Term of a Geometric Sequence Demystified

Hey everyone! Today, we're diving deep into the fascinating world of geometric sequences. We've got a cool problem to crack: figuring out the 16th term of a geometric sequence where the first term ( $a_1$ ) is 4 and the 8th term ( $a_8$ ) is a whopping -8,748. Sounds a bit intimidating, right? But trust me, we'll break it down step by step, and by the end, you'll be a geometric sequence pro!

Understanding Geometric Sequences

First things first, let's get our bearings. What exactly is a geometric sequence? Well, geometric sequences are sequences of numbers where each term is found by multiplying the previous term by a constant value. This constant value is super important; we call it the common ratio, often denoted by 'r'.

Think of it like this: you start with a number, multiply it by 'r', then multiply the result by 'r' again, and so on. Each multiplication gets you the next term in the sequence. For example, if we start with 2 and multiply by 3 each time, we get the sequence 2, 6, 18, 54, and so on. Here, the common ratio 'r' is 3.

The beauty of geometric sequences lies in their predictable pattern. This predictability allows us to develop formulas to find any term in the sequence without having to calculate all the terms before it. This is especially handy when we're dealing with terms way down the line, like the 16th term in our problem.

The General Formula

The cornerstone of working with geometric sequences is the general formula. This formula is our key to unlocking any term in the sequence, and it looks like this:

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

Where:

$a_n$ is the nth term (the term we want to find)
$a_1$ is the first term of the sequence
r is the common ratio
n is the term number (the position of the term in the sequence)

This formula might look a little daunting at first, but it's actually quite straightforward. It simply says that to find any term ( $a_n$ ), we take the first term ( $a_1$ ), multiply it by the common ratio (r) raised to the power of (n-1). This (n-1) exponent is crucial because it reflects the number of times we multiply by the common ratio to get to the nth term from the first term.

For instance, to get to the 3rd term from the first, we multiply by 'r' twice. To get to the 8th term, we multiply by 'r' seven times. This is why we have (n-1) in the exponent.

Now that we have the general formula in our toolkit, we're well-equipped to tackle our problem. But before we jump into finding the 16th term, we need to figure out the common ratio, 'r'.

Finding the Common Ratio (r)

Alright, the common ratio 'r' is the secret sauce of any geometric sequence. It dictates how the sequence grows or shrinks, and without it, we're stuck. In our problem, we don't have 'r' directly given to us, but we do have some valuable clues: the first term ( $a_1 = 4$ ) and the eighth term ( $a_8 = -8,748$ ). This is enough information to crack the code and find 'r'.

We'll use the general formula again, but this time, we'll plug in the information we know and solve for 'r'. We know $a_8 = -8,748$ , $a_1 = 4$ , and n = 8. Let's plug these values into our formula:

$-8,748 = 4 * r^(8-1)$

This simplifies to:

$-8,748 = 4 * r^7$

Now, we need to isolate $r^7$ . To do this, we'll divide both sides of the equation by 4:

$-2,187 = r^7$

Okay, we're getting closer! We now have $r^7$ isolated, but we want 'r' itself. To get rid of the exponent of 7, we need to take the 7th root of both sides. This might sound a bit intimidating, but most calculators have a root function that can handle this. The 7th root of -2,187 is -3. So, we have:

r = -3

Bingo! We've found our common ratio. It's -3, which means that each term in the sequence is obtained by multiplying the previous term by -3. The negative sign also tells us that the terms will alternate in sign (positive, negative, positive, and so on).

With 'r' in hand, we're now ready to find the 16th term. We have all the pieces of the puzzle; it's time to put them together.

Calculating the 16th Term ( $a_{16}$ )

Okay, guys, this is the moment we've been building up to! We're going to find the 16th term ( $a_{16}$ ) of our geometric sequence. We have all the information we need: the first term ( $a_1 = 4$ ), the common ratio (r = -3), and the term number (n = 16). It's time to unleash the general formula one last time!

Let's plug our values into the formula:

$a_{16} = 4 * (-3)^(16-1)$

This simplifies to:

$a_{16} = 4 * (-3)^15$

Now, we need to calculate $(-3)^15$ . This is where a calculator comes in handy, especially since we're dealing with a large exponent. $(-3)^15$ is a very large negative number: -14,348,907.

So, our equation becomes:

$a_{16} = 4 * (-14,348,907)$

Finally, we multiply 4 by -14,348,907 to get our answer:

$a_{16} = -57,395,628$

And there you have it! The 16th term of the geometric sequence is -57,395,628. That's a pretty massive number, which highlights how quickly geometric sequences can grow (or shrink, depending on the value of 'r').

Putting It All Together

Let's recap what we've done. We started with a problem: finding the 16th term of a geometric sequence given the first and eighth terms. We then:

Reviewed the definition of a geometric sequence and its general formula.
Used the given information to find the common ratio 'r'.
Plugged the values of $a_1$ , 'r', and n into the general formula to calculate $a_{16}$ .

This process demonstrates the power of the general formula and how it allows us to find any term in a geometric sequence, no matter how far down the line it is. The ability to manipulate and apply formulas like this is a crucial skill in mathematics, and it opens the door to solving a wide range of problems.

Real-World Applications and Further Exploration

Geometric sequences aren't just abstract mathematical concepts; they pop up in various real-world scenarios. For example, they can model population growth, compound interest, and the decay of radioactive substances. Understanding geometric sequences can give you insights into these phenomena and help you make predictions about the future.

Beyond the Basics

If you're feeling adventurous, there's a whole world of geometric sequence concepts to explore beyond finding individual terms. You can delve into topics like:

Geometric Series: This involves finding the sum of the terms in a geometric sequence. There are formulas to calculate the sum of a finite number of terms and even the sum of an infinite geometric series (under certain conditions).
Applications in Finance: Geometric sequences are fundamental to understanding concepts like compound interest, annuities, and mortgages.
Fractals: These fascinating geometric shapes often exhibit self-similarity based on geometric sequences.

Practice Makes Perfect

The best way to master geometric sequences is to practice solving problems. Try working through different examples, varying the given information and the term you're trying to find. Don't be afraid to make mistakes; they're valuable learning opportunities. The more you practice, the more comfortable you'll become with the concepts and the formulas.

And remember, mathematics is not just about memorizing formulas; it's about understanding the underlying principles and applying them creatively. Geometric sequences are a beautiful example of how patterns and formulas can help us make sense of the world around us. So keep exploring, keep questioning, and keep having fun with math!

Conclusion

So, guys, we successfully navigated the world of geometric sequences and found the 16th term of a sequence with a tricky common ratio. We've seen how the general formula acts as our trusty guide, allowing us to leapfrog through the sequence and pinpoint any term we desire. Remember, the key is to understand the underlying principles, practice applying the formulas, and never be afraid to tackle a challenge. Who knows what other mathematical mysteries you'll unlock next!