Geometric Progression Problem Solving Find The Number Of Terms

by Scholario Team 63 views

Hey guys! Today, let's dive into the fascinating world of geometric progressions (GPs). We've got a fun problem to solve that will help us understand how these sequences work. Our mission, should we choose to accept it, is to figure out how many terms are in a GP where the first term is 2, the last term is a whopping 1458, and the common ratio is 3. Buckle up, because we're about to unravel this mathematical mystery!

What is a Geometric Progression?

First, let's get our bearings. What exactly is a geometric progression? Well, 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." Think of it like a snowball rolling down a hill, gaining size at a consistent rate. In our case, the common ratio is 3, meaning each term is three times bigger than the one before it. So, if we start with 2, the next term is 2 * 3 = 6, then 6 * 3 = 18, and so on. The beauty of a geometric progression lies in its predictable pattern, which allows us to calculate any term in the sequence if we know the first term and the common ratio. This predictability is not just a mathematical curiosity; it has real-world applications in areas like finance (compound interest), physics (exponential decay), and even computer science (algorithms). Understanding geometric progressions is like unlocking a secret code to a wide range of phenomena, making it a valuable tool in any problem-solver's arsenal. From calculating the future value of an investment to modeling the spread of a disease, the principles of geometric progressions are surprisingly versatile and applicable to everyday situations.

Unpacking the Problem

Okay, let's break down the problem we're facing. We know the first term (a) is 2. This is our starting point, the seed from which the entire sequence grows. We also know the last term (l) is 1458. This is the final destination, the peak of our numerical journey. And, of course, we know the common ratio (r) is 3. This is the engine driving our progression, the multiplier that propels us from one term to the next. What we don't know, and what we're trying to find, is the number of terms (n) in this progression. How many steps did it take to get from 2 to 1458, multiplying by 3 each time? That's the million-dollar question, or rather, the number-of-terms question! To visualize this, imagine a staircase where the first step is 2 units high, and each subsequent step is three times the height of the previous one. We're trying to figure out how many steps it takes to reach a height of 1458 units. The key to solving this problem lies in understanding the relationship between the first term, the common ratio, the last term, and the number of terms in a geometric progression. This relationship is encapsulated in a simple yet powerful formula, which we'll explore in the next section. So, let's keep our thinking caps on and get ready to unravel the mystery of this geometric progression!

The Formula to the Rescue

Here's where the magic happens! There's a handy-dandy formula that connects all these pieces of information. The formula for the nth term (or in our case, the last term) of a geometric progression is:

l = a * r^(n-1)

Where:

  • l is the last term
  • a is the first term
  • r is the common ratio
  • n is the number of terms (the mystery we're trying to solve!)

This formula is like a decoder ring for geometric progressions. It tells us that the last term is equal to the first term multiplied by the common ratio raised to the power of (n-1). Think of it as a recipe: if you know the ingredients (a and r) and the final dish (l), you can figure out how many steps (n) it took to cook it up. In our case, we have all the ingredients except for one, and that's the number of terms. So, all we need to do is plug in the values we know and solve for n. It's like solving a puzzle where all the pieces are in place except for one, and once we find that missing piece, the whole picture becomes clear. This formula is not just a mathematical abstraction; it's a powerful tool that allows us to make predictions and solve problems in a variety of contexts. From calculating the growth of a population to determining the depreciation of an asset, the principles of geometric progressions and this formula, in particular, can provide valuable insights. So, let's put this formula to work and see how it helps us crack the code of our geometric progression!

Plugging in the Values

Now comes the fun part ā€“ substituting the values we know into the formula. We've got:

  • l = 1458
  • a = 2
  • r = 3

So, our formula becomes:

1458 = 2 * 3^(n-1)

This equation is like a mathematical sentence, and our task is to decipher it. We need to isolate 'n', the number of terms, to find its value. It's like playing a game of mathematical hide-and-seek, where 'n' is hiding behind a series of operations, and we need to carefully undo those operations to reveal its true identity. The first step is to simplify the equation by dividing both sides by 2. This gets us closer to isolating the term with 'n' in it. Then, we'll need to deal with the exponent. This is where logarithms come into play, but we'll get to that in a moment. For now, let's focus on the mechanics of substituting the values and setting up the equation. It's like laying the foundation for a building; we need to make sure the base is solid before we start constructing the higher floors. The accuracy of our final answer depends on the correctness of this initial substitution, so it's crucial to double-check our work and ensure that we've plugged in the values correctly. Once we've done that, we're well on our way to solving the problem and discovering the number of terms in our geometric progression.

Solving for 'n'

Alright, let's solve this equation step-by-step:

  1. Divide both sides by 2:

    1458 / 2 = 3^(n-1)
729 = 3^(n-1)

  2. Express 729 as a power of 3: (This is a crucial step! We need to get both sides of the equation with the same base.)

    729 = 3^6

    So, we have:

    3^6 = 3^(n-1)

  3. Since the bases are the same, the exponents must be equal: This is a fundamental property of exponents. If a^m = a^n, then m = n.

    6 = n - 1

  4. Solve for n:

    n = 6 + 1
n = 7

And there you have it! We've cracked the code! The number of terms in the geometric progression is 7. It's like reaching the summit of a mountain after a challenging climb; the view is breathtaking! Each step in this process was crucial, from dividing both sides by 2 to expressing 729 as a power of 3. The key was to recognize the underlying mathematical principles at play and apply them systematically. By understanding the properties of exponents and logarithms, we were able to transform a seemingly complex equation into a simple one that we could easily solve. This process highlights the power of mathematical reasoning and the satisfaction of arriving at a correct solution. It's not just about getting the right answer; it's about understanding the journey and the tools we used along the way. So, let's celebrate our success and move on to the next challenge, armed with the knowledge and confidence we've gained from solving this geometric progression problem!

Conclusion: The Progression Has 7 Terms

So, there you have it! The geometric progression 2, 6, 18, ..., 1458 has 7 terms. We successfully navigated the world of geometric progressions, used the formula, and solved for the unknown. High five! Understanding geometric progressions is more than just memorizing formulas; it's about grasping the underlying patterns and relationships that govern these sequences. It's about seeing the connections between different mathematical concepts and applying them creatively to solve problems. In this case, we combined our knowledge of exponents, logarithms, and algebraic manipulation to unravel the mystery of the missing number of terms. This is the essence of mathematical problem-solving: taking what you know and using it to discover what you don't know. And the more problems we solve, the more confident and skilled we become in this process. So, let's keep exploring the world of mathematics, embracing the challenges and celebrating the victories along the way. Each problem solved is a step forward on our mathematical journey, and with each step, we gain a deeper appreciation for the beauty and power of this fascinating subject.

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