Finding The Nth Term Of A Sequence: A Step-by-Step Guide

Aug 7, 2025 by Mei Lin 57 views

Decoding the Sequence: Finding the nth Term

Hey guys! Let's dive into a cool math problem today. We've got a sequence where the first term is 9, and each term after that is 4 times the previous one. Our mission? To find an equation that gives us the nth term of the sequence. Sounds like fun, right? Let's break it down step by step.

Understanding the Sequence

Okay, so the first thing we need to do is really understand the sequence. Sequences in mathematics are essentially ordered lists of numbers or other elements. They follow a specific pattern or rule. In our case, we have a geometric sequence. What's a geometric sequence, you ask? Well, it's a sequence where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio.

In our problem, the first term is 9. The next term is 4 times 9, which is 36. The term after that is 4 times 36, which is 144. And so on. So, we're multiplying by 4 each time. This means our common ratio is 4. This is super important because the common ratio plays a key role in figuring out the general formula for the sequence. To solidify our understanding, let's list out the first few terms:

Term 1: 9
Term 2: 9 * 4 = 36
Term 3: 36 * 4 = 144
Term 4: 144 * 4 = 576

See the pattern? Each term is the first term (9) multiplied by 4 a certain number of times. But how many times? That's where the term number n comes in. Think about it: for the second term, we multiply by 4 once; for the third term, we multiply by 4 twice; for the fourth term, we multiply by 4 three times. Notice that the number of times we multiply by 4 is always one less than the term number. This is a crucial insight!

Now, let's talk about the general formula for a geometric sequence. Knowing this formula will save us a lot of time and effort. The general formula is expressed as: a_n = a₁ * r^(n-1) where:

a_n represents the nth term of the sequence.
a₁ is the first term of the sequence.
r is the common ratio.
n is the term number.

This formula is a powerful tool because it allows us to find any term in the sequence without having to calculate all the preceding terms. In our specific case, we know that a₁ (the first term) is 9 and r (the common ratio) is 4. So, we can plug these values into the general formula to get an equation that gives us the nth term, which we are calling w. Let's do that!

Building the Equation

Alright, let's put our knowledge to work and build the equation that gives us w (the nth term) in terms of n. We already have the general formula for a geometric sequence: a_n = a₁ * r^(n-1). Now, we just need to substitute the values we know into this formula.

Remember, we're given that the first term (a₁) is 9 and the common ratio (r) is 4. Also, we're using w to represent the nth term (a_n). So, let's plug those values in:

w = 9 * 4^(n-1)

And there you have it! This equation gives us the nth term (w) of the sequence in terms of n. It looks pretty straightforward, right? But let's take a moment to really think about what this equation is telling us. It's saying that to find any term in the sequence, we start with 9 and multiply it by 4 raised to the power of (n-1). The exponent (n-1) makes sense because, as we discussed earlier, we multiply by 4 one less time than the term number. For instance:

To find the 2nd term (n=2), we multiply 9 by 4^(2-1) = 4¹ = 4, which gives us 36.
To find the 3rd term (n=3), we multiply 9 by 4^(3-1) = 4² = 16, which gives us 144.

See how it works? The equation perfectly captures the pattern of the sequence. This equation is the key to unlocking any term in this sequence! So, now we have a solid equation that represents this sequence. But let's make sure we really understand how it works by testing it out with a few values of n. This will not only help us confirm that our equation is correct but also deepen our understanding of the sequence itself.

Verifying the Equation

To make sure our equation is solid, let's verify the equation we found by plugging in a few values of n. This is a great way to check our work and make sure we haven't made any mistakes. Plus, it will give us a better feel for how the equation works in practice. We already know the first few terms of the sequence, so let's use those to test our equation:

w = 9 * 4^(n-1)

Let's start with n = 1 (the first term):

w = 9 * 4^(1-1)
w = 9 * 4⁰
Remember that any number raised to the power of 0 is 1, so 4⁰ = 1
w = 9 * 1
w = 9

Yep, that matches the first term we were given! Let's try n = 2 (the second term):

w = 9 * 4^(2-1)
w = 9 * 4¹
w = 9 * 4
w = 36

That's also correct! The second term is indeed 36. Let's do one more, just to be absolutely sure. Let's try n = 3 (the third term):

w = 9 * 4^(3-1)
w = 9 * 4²
w = 9 * 16
w = 144

Fantastic! Our equation works perfectly for the first three terms. We can be pretty confident that it's the correct equation for this sequence. By plugging in values for n, we've not only verified our equation but also reinforced our understanding of how the sequence grows. Each term is simply 9 multiplied by a power of 4, where the power is one less than the term number. This kind of verification is a super useful strategy in math. Whenever you find a formula or equation, it's always a good idea to test it out with some specific examples. This helps you catch any errors and also deepens your understanding of the concepts involved. Now that we've successfully built and verified the equation, we've essentially solved the problem! But let's take a step back and think about the broader implications of what we've done.

Conclusion

So, there you have it! We successfully navigated the world of sequences and found an equation that describes the nth term of our given sequence. By understanding the concept of a geometric sequence, the common ratio, and the general formula, we were able to build the equation w = 9 * 4^(n-1). We then verified this equation by plugging in a few values of n and confirming that it matched the terms of the sequence. This process demonstrates the power of mathematical reasoning and problem-solving.

This problem wasn't just about finding an answer; it was about understanding the underlying principles and applying them in a logical way. We started by carefully analyzing the given information, then identified the pattern of the sequence, and finally used the general formula for a geometric sequence to derive the equation. This is a generalizable approach, meaning it can be applied to solve similar problems with different numbers and sequences. The key takeaway here is that math isn't just about memorizing formulas; it's about understanding the concepts and developing the skills to apply them in different situations.

If you encounter a similar problem in the future, remember the steps we took here: 1) understand the sequence, 2) identify the pattern, 3) use the general formula, and 4) verify your answer. With practice, you'll become a pro at solving these kinds of problems! I hope this explanation was helpful and insightful. Keep exploring the fascinating world of mathematics, guys!