Geometric Sequence: Find First 3 Terms | A_5=25, R=5

by Andrew McMorgan 53 views

Hey Plastik Magazine readers! Let's dive into a cool math problem today. We're going to explore how to find the first three terms of a geometric sequence. This might sound a bit intimidating, but trust me, we'll break it down and make it super easy to understand. We've got a specific scenario: the fifth term ( extit{a_5}) of our sequence is 25, and the common ratio ( extit{r}) is 5. So, how do we backtrack and figure out those initial terms? Let’s get started!

Understanding Geometric Sequences

Before we jump into solving the problem, let's quickly recap what a geometric sequence actually is. In a nutshell, it’s 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, often denoted by extitr}. Think of it like this you start with a number, and then you multiply it by extit{r to get the next number, and so on. The general form of a geometric sequence can be written as:

  • a, ar, ar^2, ar^3, ar^4, ...

Where:

  •   extit{a} is the first term

  •   extit{r} is the common ratio

  •   extit{ar^(n-1)} is the nth term

For example, if our first term ( extit{a}) is 2 and the common ratio ( extit{r}) is 3, the sequence would look like this: 2, 6, 18, 54, and so on. Each term is simply the previous term multiplied by 3. Grasping this basic concept is key to tackling our problem. We need to work backwards from the fifth term to uncover the first three, and understanding the relationship between terms in a geometric sequence is crucial for that. So, now that we've refreshed our understanding of geometric sequences, let's roll up our sleeves and get to the fun part: solving the problem!

Setting Up the Problem

Okay, guys, let's get down to business! We know two crucial pieces of information: the fifth term ( extit{a_5}) is 25, and the common ratio ( extit{r}) is 5. Our mission, should we choose to accept it (and we do!), is to find the first three terms of this geometric sequence. To kick things off, we need to remember the formula for the nth term of a geometric sequence. As we mentioned earlier, it's given by:

  • a_n = a * r^(n-1)

Where:

  •   extit{a_n} is the nth term

  •   extit{a} is the first term (which we're trying to find!)

  •   extit{r} is the common ratio (which we know is 5)

  •   extit{n} is the term number

In our specific case, we know that extit{a_5} = 25 and extit{r} = 5. So, we can plug these values into the formula:

  • 25 = a * 5^(5-1)
  • 25 = a * 5^4

Now we have an equation with one unknown – extit{a}, the first term. This is fantastic because once we find extit{a}, we're well on our way to figuring out the other two terms. Think of it like setting up a treasure map. We've got a landmark (the fifth term), and we know the direction to the treasure (the common ratio). Now we just need to calculate the distance (the first term). So, let's put on our thinking caps and solve for extit{a}! The next step involves some simple algebra, and we'll have our first piece of the puzzle in no time.

Solving for the First Term (a)

Alright, let's get our algebra muscles flexing! We've got the equation 25 = a * 5^4, and our goal is to isolate extit{a}. This basically means getting extit{a} all by itself on one side of the equation. The first thing we need to do is figure out what 5^4 is. If you punch that into your calculator (or do it the old-fashioned way!), you'll find that 5^4 = 625. So, our equation now looks like this:

  • 25 = a * 625

Now, to get extit{a} by itself, we need to undo the multiplication. We do this by dividing both sides of the equation by 625. This is a fundamental principle of algebra – whatever you do to one side of the equation, you have to do to the other to keep things balanced. So, let's divide both sides by 625:

  • 25 / 625 = (a * 625) / 625

This simplifies to:

  • 25 / 625 = a

Now we just need to simplify the fraction 25/625. Both 25 and 625 are divisible by 25, so let's divide both the numerator and the denominator by 25:

  • (25 / 25) / (625 / 25) = a
  • 1 / 25 = a

Voila! We've found our first term. The first term ( extit{a}) of the geometric sequence is 1/25. That wasn't so bad, was it? Now that we've got the first piece of the puzzle, finding the next two terms will be a breeze. We've successfully navigated the algebraic waters and emerged victorious with our value for extit{a}. Let's keep this momentum going and find the second and third terms!

Finding the Second and Third Terms

Awesome! We've nailed down the first term ( extita} = 1/25). Now comes the fun part using the common ratio ( extit{r = 5) to find the second and third terms. Remember, in a geometric sequence, each term is obtained by multiplying the previous term by the common ratio. So, to find the second term ( extit{a_2}), we simply multiply the first term ( extit{a_1}) by the common ratio ( extit{r}):

  • a_2 = a_1 * r
  • a_2 = (1/25) * 5

To multiply a fraction by a whole number, we can rewrite the whole number as a fraction with a denominator of 1:

  • a_2 = (1/25) * (5/1)

Now we multiply the numerators and the denominators:

  • a_2 = (1 * 5) / (25 * 1)
  • a_2 = 5 / 25

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

  • a_2 = (5 / 5) / (25 / 5)
  • a_2 = 1 / 5

So, the second term of the sequence is 1/5. We're on a roll! Now, let's find the third term ( extita_3}). We can use the same method multiply the second term ( extit{a_2) by the common ratio ( extit{r}):

  • a_3 = a_2 * r
  • a_3 = (1/5) * 5

Again, we can rewrite 5 as 5/1:

  • a_3 = (1/5) * (5/1)

Multiply the numerators and the denominators:

  • a_3 = (1 * 5) / (5 * 1)
  • a_3 = 5 / 5

This simplifies to:

  • a_3 = 1

Boom! The third term of the sequence is 1. We've successfully found the first three terms of our geometric sequence. High fives all around! We started with just the fifth term and the common ratio, and we worked our way back to the beginning. Now, let's put it all together and see the sequence in its full glory.

The First Three Terms Revealed

Drumroll, please! We've cracked the code and found the first three terms of the geometric sequence. Let's recap what we discovered:

  • The first term ( extit{a_1}) is 1/25.
  • The second term ( extit{a_2}) is 1/5.
  • The third term ( extit{a_3}) is 1.

So, the first three terms of the geometric sequence are 1/25, 1/5, and 1. Pretty cool, right? We started with a bit of a puzzle, knowing only the fifth term and the common ratio, and we used our knowledge of geometric sequences and some basic algebra to unravel the mystery. This is what makes math so fascinating – it's like detective work with numbers! We followed the clues, applied the formulas, and arrived at the solution. This exercise demonstrates the power of understanding the fundamental principles of geometric sequences. By knowing the formula for the nth term and how the common ratio works, we can solve a wide range of problems. Whether you're dealing with simple sequences or more complex scenarios, the core concepts remain the same. So, keep practicing, keep exploring, and keep those math muscles strong! Who knows what mathematical mysteries you'll solve next?

Wrapping It Up

Alright, Plastik Magazine crew, we've reached the end of our geometric sequence adventure! We successfully found the first three terms of a sequence where the fifth term was 25 and the common ratio was 5. We revisited the definition of a geometric sequence, dusted off the formula for the nth term, and put our algebraic skills to the test. We learned that by understanding the relationship between terms in a geometric sequence, we can work both forwards and backwards to solve problems. Whether we're finding a specific term or unraveling the initial terms, the fundamental principles remain our trusty guides. Remember, the key takeaways from today's exploration are:

  1. The formula for the nth term of a geometric sequence: a_n = a * r^(n-1)
  2. How to use the common ratio to move between terms in a sequence.
  3. The importance of algebraic manipulation in solving for unknowns.

So, the next time you encounter a geometric sequence problem, don't sweat it! Break it down, identify what you know, and apply the principles we've discussed. Math might seem like a daunting subject at times, but with a little practice and the right approach, it can be both accessible and rewarding. And who knows, maybe you'll even start seeing geometric sequences in the world around you – from the growth of populations to the decay of radioactive substances. Math is everywhere, and the more we understand it, the more we can appreciate the intricate patterns that govern our universe. Until next time, keep exploring, keep learning, and keep those mathematical gears turning! Peace out!