Finding The 3rd Term Of A Sequence: A Step-by-Step Guide

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Hey guys! Ever stumbled upon a sequence and wondered how to find a specific term? Sequences might seem intimidating at first, but trust me, they're pretty straightforward once you get the hang of it. Today, we're diving into a specific type of sequence and figuring out how to nail down its 3rd term. Let's break it down together!

Understanding the Sequence

So, we've got this sequence defined by two key pieces of information:

  • The first term: a(1) = 20. This tells us where the sequence kicks off.
  • The recursive formula: a(n) = a(n-1) * (3/2). This is the juicy part! It tells us how to get to the next term based on the one before it. In simpler terms, to find any term, you multiply the previous term by 3/2. This type of sequence, where each term is found by multiplying the previous term by a constant value, is called a geometric sequence. The constant value (in this case, 3/2) is known as the common ratio.

Understanding this formula is crucial. It's the engine that drives our sequence forward. Think of it like a set of instructions: "Hey, to get the next number, just take the current one and multiply it by 3/2." It's a chain reaction, each term building upon the last. This recursive nature is what makes these sequences so interesting. You can start with a simple rule and generate a whole string of numbers. Imagine the possibilities!

The beauty of recursive formulas is that they allow us to define sequences without explicitly listing out every single term. Instead, we define the starting point and then provide a rule for how to move from one term to the next. It's like setting up dominoes – you only need to set up the first domino and know how they're spaced apart, and the rest will fall into place. In this case, a(1) = 20 is our first domino, and a(n) = a(n-1) * (3/2) is the spacing rule.

Now, let's think about what we need to do to find the third term. The recursive formula tells us how to get the next term, so to get to the third term, we'll need to use the formula twice. First, we'll use it to find the second term (a(2)), and then we'll use it again with a(2) to find the third term (a(3)). It's like climbing a ladder – each step (each application of the formula) takes us closer to our goal.

Calculating the Second Term (a(2))

Okay, let's put that recursive formula to work! We want to find a(2), the second term in the sequence. Our formula says a(n) = a(n-1) * (3/2). To find a(2), we simply substitute n = 2 into the formula:

a(2) = a(2-1) * (3/2)

This simplifies to:

a(2) = a(1) * (3/2)

Remember, we know that a(1) = 20. So, we can plug that in:

a(2) = 20 * (3/2)

Now, it's just a matter of doing the math. 20 multiplied by 3/2 is the same as (20 * 3) / 2, which is 60 / 2. And that gives us:

a(2) = 30

So, the second term in our sequence is 30! We've successfully used the recursive formula to take a step forward. Think of it like leveling up in a game – we've used our initial knowledge (a(1) = 20) and our core mechanic (the recursive formula) to progress to the next stage. But we're not done yet! We need to find the third term, and we'll use the same principle to get there.

What we've done here is a classic example of how recursion works. We've defined a problem (finding a term in a sequence) in terms of a smaller version of the same problem (finding the previous term). This self-referential approach is powerful because it allows us to solve complex problems by breaking them down into smaller, more manageable steps. In computer science, recursion is a fundamental concept used in many algorithms and data structures. And here, in the world of sequences, it provides us with a neat way to generate terms one after the other.

Finding the Third Term (a(3))

Alright, we've conquered the second term, and now it's time to set our sights on the third! We'll use the same recursive formula, a(n) = a(n-1) * (3/2), but this time, we're aiming for a(3). So, we substitute n = 3 into the formula:

a(3) = a(3-1) * (3/2)

Which simplifies to:

a(3) = a(2) * (3/2)

This is where our previous victory comes in handy! We already know that a(2) = 30. We can now plug that value into our equation:

a(3) = 30 * (3/2)

Time for some more arithmetic! 30 multiplied by 3/2 is the same as (30 * 3) / 2, which equals 90 / 2. And that gives us:

a(3) = 45

Boom! We've found it! The third term in the sequence is 45. We've successfully navigated the sequence, step by step, using the power of the recursive formula. It's like we've built a bridge across the sequence, each term serving as a stepping stone to the next. And we've reached our destination – the third term.

This process highlights the elegance of recursive definitions. By defining each term in relation to the one before it, we can efficiently calculate any term in the sequence, provided we have a starting point. We didn't need to list out every term leading up to the third; we simply used the formula and the previously calculated term to jump directly to our target. This efficiency becomes even more apparent when dealing with sequences where we need to find terms much further down the line, like the 10th or 100th term. Imagine trying to calculate the 100th term without a recursive formula – it would be a daunting task! But with this handy tool, we can tackle those challenges with ease.

Wrapping It Up

So, there you have it! We've successfully found the 3rd term in the sequence defined by a(1) = 20 and a(n) = a(n-1) * (3/2). We walked through the process step-by-step:

  1. Understood the sequence: We identified it as a geometric sequence with a common ratio of 3/2 and a starting term of 20.
  2. Calculated the second term: We used the recursive formula to find a(2) = 30.
  3. Found the third term: We used the recursive formula again, along with our value for a(2), to determine that a(3) = 45.

This example demonstrates the power of recursive formulas in defining and working with sequences. It might seem a bit abstract at first, but once you break it down, it's just a matter of applying the formula repeatedly. Keep practicing, and you'll be a sequence-solving pro in no time! Remember, math is all about building upon your knowledge, each concept laying the foundation for the next. And sequences are a fundamental building block in many areas of mathematics and computer science. So, mastering them is a fantastic step towards unlocking even more mathematical mysteries!

If you guys have any questions or want to explore other types of sequences, feel free to ask! Keep exploring and keep learning!