Geometric Progression: Find R And A4 Given T1 And T6

by ADMIN 53 views

Hey guys! Today, we're diving into the fascinating world of geometric progressions. We've got a cool problem to solve that involves finding the common ratio (r) and a specific term (a4) in a geometric sequence, given some initial information. If you're scratching your head about geometric progressions, don't worry! We'll break it down step-by-step so it's super easy to understand. So, let’s get started and unravel this mathematical mystery together!

Understanding Geometric Progressions

First things first, let's make sure we're all on the same page about what a geometric progression actually is. Think of it as a sequence of numbers where each term is found by multiplying the previous term by a constant value. This constant value is what we call the common ratio, denoted by r. It's this r that dictates how the sequence grows (or shrinks!).

To illustrate, imagine we start with a number, say 2. If our common ratio r is 3, the geometric progression would look like this: 2, 6, 18, 54, and so on. See how each term is simply the previous term multiplied by 3? That's the magic of geometric progressions!

The General Formula

The backbone of understanding and working with geometric progressions is the general formula. This formula allows us to find any term in the sequence without having to calculate all the terms before it. It's a real time-saver, trust me!

The general formula for the nth term (tn) of a geometric progression is given by:

tn = t1 * r*^(n-1)

Where:

  • tn is the nth term we want to find.
  • t1 is the first term of the sequence.
  • r is the common ratio.
  • n is the position of the term in the sequence.

This formula is our trusty tool for solving all sorts of geometric progression problems. We'll be using it extensively in this article, so make sure you have it down! It might seem a bit intimidating at first, but once you use it a few times, you'll see how straightforward it is. It basically says, “To find any term, take the first term and multiply it by the common ratio raised to the power of one less than the term number.” Simple, right?

Why Geometric Progressions Matter

Now, you might be wondering, “Okay, this is interesting, but why should I care about geometric progressions?” Well, the truth is, geometric progressions pop up in all sorts of real-world scenarios! They're not just some abstract mathematical concept; they have practical applications that make them incredibly useful.

Here are a few examples of where you might encounter geometric progressions:

  • Compound Interest: When you invest money and it earns compound interest, the amount grows geometrically. Each year, the interest is added to the principal, and the next year's interest is calculated on the new, larger amount. This leads to exponential growth, which is a key characteristic of geometric progressions.
  • Population Growth: In certain situations, populations can grow geometrically. If a population doubles every year, for example, the population sizes will form a geometric progression.
  • Radioactive Decay: On the flip side, radioactive decay follows a geometric pattern, but in reverse. The amount of radioactive material decreases by a constant factor over time, creating a decaying geometric sequence.
  • Computer Science: Geometric progressions are used in algorithms and data structures, particularly in scenarios involving repeated halving or doubling of values.

These are just a few examples, but they highlight how versatile and relevant geometric progressions are. Understanding them gives you a powerful tool for analyzing and predicting patterns in a variety of fields. Plus, they're kinda fun to work with once you get the hang of it! So, let's get back to our problem and put this knowledge into action.

Problem Setup: Decoding the Given Information

Alright, let's dive into the problem at hand. We're given some crucial information about a geometric progression, and our mission is to find the common ratio (r) and the fourth term (a4). Think of it like being a detective – we have clues, and we need to use them to solve the mystery!

Here's what we know:

  • The first term, t1, is 2.
  • The sixth term, t6, is 64.

That's it! It might not seem like much, but with our understanding of geometric progressions and the general formula, we can work wonders. The key is to translate this information into mathematical equations that we can then solve.

The first piece of information, t1 = 2, is pretty straightforward. It tells us where the sequence starts. The second piece, t6 = 64, gives us a term further down the line. This is where the general formula comes in handy. We can use the fact that t6 is 64 to create an equation involving r, the common ratio, which is what we're trying to find.

So, let's apply the general formula to t6. Remember the formula:

tn = t1 * r*^(n-1)

In our case, n = 6 (since we're dealing with the sixth term), t6 = 64, and t1 = 2. Plugging these values into the formula, we get:

64 = 2 * r^(6-1)

This simplifies to:

64 = 2 * r^5

Boom! We've got an equation that relates the common ratio r to known values. Now, it's just a matter of solving for r. Think of this equation as a puzzle – we need to isolate r to uncover its value. Don't worry, we'll take it step-by-step.

But before we jump into solving, let's take a moment to appreciate what we've done. We've successfully translated the problem's given information into a mathematical equation. This is a crucial skill in problem-solving, not just in math but in all areas of life. Being able to break down a complex problem into smaller, manageable pieces is key to finding solutions. So, give yourself a pat on the back for making it this far! Now, let's tackle that equation and find r.

Finding the Common Ratio (r): Solving the Equation

Okay, we've arrived at a pivotal point in our journey – solving for the common ratio, r. Remember the equation we derived:

64 = 2 * r^5

Our goal here is to isolate r on one side of the equation. To do this, we'll use some basic algebraic techniques. Don't worry, it's not as scary as it might sound! We'll take it one step at a time.

First, let's get rid of that 2 that's multiplying the r^5. To do this, we'll divide both sides of the equation by 2. This is a fundamental principle of algebra: whatever you do to one side of the equation, you must do to the other to maintain the balance.

Dividing both sides by 2, we get:

64 / 2 = (2 * r^5) / 2

This simplifies to:

32 = r^5

Great! We've made progress. Now we have r^5 isolated on one side. But we want r, not r^5. So, how do we get rid of that exponent of 5? This is where we need to use the concept of roots.

To undo the power of 5, we need to take the fifth root of both sides of the equation. Just like dividing both sides, taking the root of both sides maintains the equality. The fifth root of a number is the value that, when multiplied by itself five times, equals the original number.

So, we take the fifth root of both sides:

∛(32) = ∛(r^5)

The fifth root of 32 is 2 (because 2 * 2 * 2 * 2 * 2 = 32), and the fifth root of r^5 is simply r. Therefore, our equation simplifies to:

r = 2

Eureka! We've found the common ratio! The value of r is 2. This means that each term in our geometric progression is double the previous term. Isn't it satisfying when a plan comes together? We started with a seemingly complex equation, and by applying basic algebraic principles, we've successfully unearthed the value of r. Give yourself another pat on the back – you're doing awesome!

Now that we've conquered the common ratio, our next task is to find a4, the fourth term of the sequence. We have all the pieces of the puzzle now, so let's put them together and complete our mission!

Calculating a4: Finding the Fourth Term

Fantastic! We've successfully determined that the common ratio, r, is 2. Now, let's use this knowledge to find a4, the fourth term of our geometric progression. Remember, a4 is just another way of saying t4, the fourth term. It’s like calling a friend by their nickname – same person, different name!

To find a4, we'll once again turn to our trusty general formula for geometric progressions:

tn = t1 * r*^(n-1)

This time, we want to find t4, so n = 4. We already know that t1 (the first term) is 2, and we've just discovered that r (the common ratio) is also 2. Plugging these values into the formula, we get:

t4 = 2 * 2^(4-1)

Now, let's simplify this step-by-step. First, we deal with the exponent:

4 - 1 = 3

So our equation becomes:

t4 = 2 * 2^3

Next, we calculate 2^3, which is 2 * 2 * 2 = 8. Our equation now looks like this:

t4 = 2 * 8

Finally, we multiply 2 by 8 to get:

t4 = 16

And there we have it! We've found that a4, the fourth term of the geometric progression, is 16. Hooray! We've successfully completed our mission. We set out to find the common ratio and the fourth term, and we've done exactly that. By understanding the principles of geometric progressions and applying the general formula, we've cracked the code.

Putting It All Together

Let's take a moment to appreciate the journey we've been on. We started with a problem that might have seemed a bit daunting at first, but by breaking it down into smaller steps and applying our knowledge, we've emerged victorious. We've not only found the answers but also strengthened our understanding of geometric progressions along the way.

To recap, here's what we found:

  • The common ratio, r, is 2.
  • The fourth term, a4, is 16.

This means our geometric progression starts with 2, and each term is double the previous term. So, the first few terms would be: 2, 4, 8, 16 (which is a4), and so on.

Conclusion: Mastering Geometric Progressions

Awesome job, guys! You've successfully navigated a geometric progression problem, found the common ratio, and calculated a specific term. You've shown that you can tackle mathematical challenges with confidence and skill. Remember, the key to mastering any mathematical concept is to break it down into smaller, manageable steps and practice regularly.

We started by understanding the fundamentals of geometric progressions, including the general formula. We then applied this knowledge to a specific problem, translating the given information into mathematical equations. By solving these equations, we unearthed the common ratio and the fourth term. It's been a journey of discovery and problem-solving, and you've aced it!

Geometric progressions are a powerful tool in mathematics and have applications in various real-world scenarios, from finance to population growth. By understanding them, you're equipping yourself with a valuable skill that will serve you well in many areas of life.

So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is vast and fascinating, and there's always something new to learn. And remember, the more you practice, the more comfortable and confident you'll become. You've got this!

If you enjoyed this exploration of geometric progressions, be sure to check out other mathematical concepts and problems. There's a whole universe of mathematical wonders waiting to be discovered. Keep up the great work, and happy problem-solving!