Explicit Formula: Decoding The Sequence 2, 10, 50, 250, 1250

by Mei Lin 61 views

Hey there, math enthusiasts! Ever stumbled upon a sequence of numbers and felt the urge to crack its code? Well, today we're diving headfirst into a fascinating sequence: 2, 10, 50, 250, 1250, and beyond. Our mission? To uncover the explicit formula that governs this numerical pattern. This means we're looking for a neat little equation that will directly tell us the value of any term in the sequence, just by plugging in its position. So, grab your thinking caps, and let's get started!

Spotting the Pattern: A Key to Unlocking the Formula

Before we jump into formulas and equations, let's take a moment to appreciate the sequence itself. What do you notice about the numbers? Do they seem to be increasing by a constant amount? Or perhaps there's a different kind of relationship at play? This initial observation is crucial, guys, because it's the compass that will guide us towards the correct formula. Looking at the sequence 2, 10, 50, 250, 1250, we can quickly see that the numbers are growing quite rapidly. This suggests that we're dealing with a geometric sequence, where each term is obtained by multiplying the previous term by a constant value, known as the common ratio.

To confirm this suspicion, let's calculate the ratio between consecutive terms. We'll divide the second term (10) by the first term (2), which gives us 5. Let's try another pair: 50 divided by 10 also equals 5. And 250 divided by 50? You guessed it – 5! This consistent ratio of 5 is our smoking gun, confirming that we are indeed working with a geometric sequence. Understanding this common ratio is paramount, as it forms the foundation of our explicit formula. Now that we've identified the sequence as geometric and found the common ratio, we're well on our way to unlocking the explicit formula. The next step involves recalling the general form of an explicit formula for geometric sequences and tailoring it to fit our specific case.

The General Form: Our Template for Success

Okay, so we know we're dealing with a geometric sequence. That's fantastic! But how do we translate this knowledge into a concrete formula? The answer lies in the general form of an explicit formula for geometric sequences. This general form acts as a template, a blueprint that we can adapt to any geometric sequence, regardless of its specific numbers. The general form looks like this:

a_n = a_1 * r^(n-1)

Where:

  • a_n represents the nth term in the sequence (the term we want to find).
  • a_1 represents the first term in the sequence.
  • r represents the common ratio (the value we multiply by to get the next term).
  • n represents the position of the term in the sequence (e.g., 1 for the first term, 2 for the second term, and so on).

This formula might look a bit intimidating at first, but trust me, it's incredibly powerful. It essentially says that any term in a geometric sequence is equal to the first term multiplied by the common ratio raised to the power of (n-1). The (n-1) part is crucial because it ensures that the first term (when n=1) is multiplied by r^(1-1) = r^0 = 1, giving us the first term itself. Now that we have this general form in our toolkit, we can move on to the exciting part: plugging in the specific values from our sequence and crafting our very own explicit formula.

Plugging in the Values: Crafting Our Specific Formula

Alright, we've got the general formula for geometric sequences down, and we've identified our sequence as geometric with a common ratio of 5. Now it's time to put the pieces together and create the explicit formula specific to our sequence: 2, 10, 50, 250, 1250, ... Remember the general form:

a_n = a_1 * r^(n-1)

We need to identify a_1 (the first term) and r (the common ratio) from our sequence. Looking at the sequence, the first term, a_1, is clearly 2. And we already figured out that the common ratio, r, is 5. So, let's substitute these values into the general formula:

a_n = 2 * 5^(n-1)

Voila! We've crafted our explicit formula. This formula, a_n = 2 * 5^(n-1), is the key to unlocking any term in the sequence. If we want to find the 10th term, we simply plug in n = 10. If we want the 100th term, we plug in n = 100. It's that simple! This formula elegantly captures the essence of the sequence, expressing the relationship between the term number and the term's value. Now, let's put this formula to the test and see if it truly holds up.

Testing the Formula: Does It Hold Up?

We've derived our explicit formula, a_n = 2 * 5^(n-1), but before we declare victory, it's crucial to test its accuracy. We need to make sure that this formula actually produces the correct terms for our sequence. A simple way to do this is to plug in a few values of n (the term number) and see if the formula spits out the corresponding terms in the sequence. Let's start with the basics:

  • n = 1 (First term): a_1 = 2 * 5^(1-1) = 2 * 5^0 = 2 * 1 = 2. This matches the first term in our sequence!
  • n = 2 (Second term): a_2 = 2 * 5^(2-1) = 2 * 5^1 = 2 * 5 = 10. This matches the second term!
  • n = 3 (Third term): a_3 = 2 * 5^(3-1) = 2 * 5^2 = 2 * 25 = 50. This matches the third term!

So far, so good! Our formula is holding up perfectly. We've successfully verified it for the first three terms. For added confidence, let's try one more:

  • n = 4 (Fourth term): a_4 = 2 * 5^(4-1) = 2 * 5^3 = 2 * 125 = 250. This matches the fourth term!

Excellent! The formula consistently generates the correct terms in the sequence. This rigorous testing gives us a high degree of confidence that our formula is indeed the explicit formula for the sequence 2, 10, 50, 250, 1250, ... Now that we've confirmed our formula, let's take a look at the answer choices provided and see which one matches our result.

Matching the Formula: Finding the Correct Answer Choice

We've successfully derived and tested our explicit formula: a_n = 2 * 5^(n-1). Now comes the final step: matching our formula to the answer choices provided. This is a critical step to ensure we select the correct option. Let's revisit the answer choices:

A. a_n = 2 + (5)^n B. a_n = 2(5)^n C. a_n = 2(5)^(n-1) D. a_n = 5(2)^(n-1)

By carefully comparing our formula, a_n = 2 * 5^(n-1), with the answer choices, we can clearly see that option C, a_n = 2(5)^(n-1), is an exact match. The other options differ significantly from our derived formula. Option A involves addition instead of multiplication, and options B and D have different exponents or base values. Therefore, we can confidently conclude that option C is the correct answer. We've not only found the explicit formula but also successfully identified it within the given choices. Give yourself a pat on the back, guys; we've cracked the code!

Conclusion: The Power of Explicit Formulas

In this mathematical adventure, we embarked on a journey to uncover the explicit formula for the sequence 2, 10, 50, 250, 1250, ... We started by observing the pattern, recognizing it as a geometric sequence with a common ratio of 5. We then leveraged the general form of an explicit formula for geometric sequences, a_n = a_1 * r^(n-1), as our template. By plugging in the specific values from our sequence (a_1 = 2 and r = 5), we crafted our own formula: a_n = 2 * 5^(n-1). We didn't stop there; we rigorously tested our formula, ensuring its accuracy by verifying it against the known terms of the sequence. Finally, we matched our formula to the answer choices and confidently selected the correct option, C. Throughout this process, we've not only solved a specific problem but also gained a deeper appreciation for the power and elegance of explicit formulas. These formulas provide a concise and powerful way to represent sequences, allowing us to calculate any term directly, without having to compute all the preceding terms. So, the next time you encounter a sequence, remember the tools and techniques we've explored today, and you'll be well-equipped to unlock its secrets!