Geometric Progression: Find The Number Of Terms

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Geometric Progression: Find the Number of Terms

Hey guys! Today, we're diving into a super common and useful math concept: geometric progressions, or GPs. Specifically, we're going to figure out how to find the number of terms in a GP when we know the first term, the last term, and the common ratio. It's like uncovering a mathematical secret! Let's break it down step-by-step, so you can totally ace any similar problem that comes your way.

Understanding Geometric Progressions

Before we jump into solving the problem, let's make sure we're all on the same page about what a geometric progression actually is. A geometric progression (GP) is a sequence of numbers where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio, often denoted as 'r'.

Example Time! If you start with the number 2 and the common ratio is 3, the GP would look like this: 2, 6, 18, 54, and so on. Notice how each term is three times the previous one? That's the magic of a GP!

The general form of a GP is usually written as: a, ar, ar², ar³, ..., ar^(n-1), where:

  • 'a' is the first term,
  • 'r' is the common ratio, and
  • 'n' is the number of terms in the GP.

Understanding this general form is super important because it gives us the tools to solve all sorts of GP-related problems. In our case, it's going to help us find out how many terms are in our specific GP.

Problem Statement

Okay, let's get back to the problem at hand. We're given a geometric progression (GP) where:

  • The first term (a) is 1.
  • The last term is 243.
  • The common ratio (r) is 3.

Our mission, should we choose to accept it (and we totally do!), is to find the total number of terms (n) in this GP. The possible answers are: A) 4 terms, B) 5 terms, C) 6 terms, and D) 7 terms. So, how do we crack this?

Solving for the Number of Terms

Here's where the general form of a GP comes to our rescue. We know that the nth term (the last term) of a GP can be expressed as: a * r^(n-1).

In our problem, we know the last term is 243, the first term (a) is 1, and the common ratio (r) is 3. So we can set up the equation like this:

1 * 3^(n-1) = 243

Since multiplying by 1 doesn't change anything, we can simplify this to:

3^(n-1) = 243

Now, we need to express 243 as a power of 3. Let's see... 3 * 3 = 9, 9 * 3 = 27, 27 * 3 = 81, and 81 * 3 = 243. So, 243 is equal to 3 raised to the power of 5 (3^5).

Therefore, our equation becomes:

3^(n-1) = 3^5

When the bases are the same (in this case, both sides have a base of 3), we can simply equate the exponents:

n - 1 = 5

Now, it's just a simple matter of solving for 'n'. Add 1 to both sides of the equation:

n = 5 + 1

n = 6

So, the total number of terms in the geometric progression is 6. That means the correct answer is C) 6 terms.

Verifying the Solution

Just to be super sure, let's list out the terms of the GP to verify our answer:

  1. First term: 1
  2. Second term: 1 * 3 = 3
  3. Third term: 3 * 3 = 9
  4. Fourth term: 9 * 3 = 27
  5. Fifth term: 27 * 3 = 81
  6. Sixth term: 81 * 3 = 243

Yep, that confirms it! There are indeed 6 terms in the GP. We nailed it!

Why Other Options are Incorrect

Let's quickly look at why the other options are wrong. This can help solidify our understanding and prevent us from making similar mistakes in the future.

  • A) 4 terms: If there were only 4 terms, the GP would be 1, 3, 9, 27. The last term would be 27, not 243.
  • B) 5 terms: If there were 5 terms, the GP would be 1, 3, 9, 27, 81. The last term would be 81, not 243.
  • D) 7 terms: If there were 7 terms, the GP would be 1, 3, 9, 27, 81, 243, 729. The last term would be 729, not 243, and we were given that 243 is the last term.

As you can see, only 6 terms allow the GP to start at 1, have a common ratio of 3, and end at 243.

Key Takeaways

Alright, let's recap the key takeaways from this problem:

  1. Geometric Progression (GP): A sequence where each term is multiplied by a constant common ratio.
  2. General Form of a GP: a, ar, ar², ar³, ..., ar^(n-1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
  3. Finding the Number of Terms: Use the formula for the nth term (a * r^(n-1)) and solve for 'n'.
  4. Verification: Always verify your solution by listing out the terms of the GP to make sure it matches the given information.

Practice Makes Perfect

The best way to master geometric progressions is to practice, practice, practice! Try solving similar problems with different first terms, common ratios, and last terms. You can even create your own GP problems to challenge yourself.

Here's a practice problem for you:

What is the number of terms in a GP that starts with 2, ends with 162, and has a common ratio of 3?

Give it a shot, and let me know your answer in the comments below! Keep practicing, and you'll become a GP pro in no time!

Conclusion

So, there you have it! We've successfully navigated the world of geometric progressions and figured out how to find the number of terms when given the first term, last term, and common ratio. Remember the key concepts, practice regularly, and you'll be well-equipped to tackle any GP problem that comes your way.

Math can be fun and rewarding, especially when you break it down into manageable steps. Keep exploring, keep learning, and never stop asking questions. You've got this!