Geometric Sequence Formula: Find The Nth Term (a₁=3, R=2)

Understanding geometric sequences is a fundamental concept in mathematics. In this article, we'll break down the process of finding the formula for the nth term of a geometric sequence, specifically when the first term (a₁) is 3 and the common ratio (r) is 2. We'll explore the general formula, apply it to our specific case, and discuss why the correct answer is B. a_n = 3 ⋅ 2^n-1.

What is a Geometric Sequence?

Before diving into the formula, let's quickly recap what a geometric sequence is. A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant value called the common ratio (r). Think of it like exponential growth or decay. For example, the sequence 2, 4, 8, 16... is a geometric sequence with a common ratio of 2 (each term is twice the previous term).

Key Components of a Geometric Sequence

To understand the formula, we need to identify the key components of a geometric sequence:

• a₁: This represents the first term of the sequence. It's our starting point.
• r: This is the common ratio. It's the constant value we multiply by to get from one term to the next.
• n: This represents the term number we want to find. For example, if we want the 5th term, n would be 5.
• a_n: This represents the nth term itself – the value of the term at position n in the sequence.

The General Formula for the nth Term

The beauty of mathematics lies in its ability to generalize concepts into formulas. For geometric sequences, there's a handy formula that allows us to calculate any term in the sequence directly, without having to list out all the preceding terms. This formula is:

a_n = a₁ ⋅ r^n-1

Let's break down what this formula tells us:

• To find the nth term (a_n), we start with the first term (a₁).
• We multiply the first term by the common ratio (r) raised to the power of (n-1). This exponent, (n-1), is crucial because it reflects the number of times we need to multiply by the common ratio to reach the nth term.

For instance, if we want to find the 3rd term, we've multiplied by the common ratio twice (from the 1st to the 2nd, and then from the 2nd to the 3rd). That's why we use n-1 as the exponent.

Applying the Formula to Our Specific Case (a₁=3, r=2)

Now, let's apply this general formula to the specific scenario given in the question: a geometric sequence where a₁ = 3 (the first term is 3) and r = 2 (the common ratio is 2). Our goal is to find the formula that represents the nth term of this particular sequence.

We simply substitute these values into the general formula:

a_n = 3 ⋅ 2^n-1*

And there we have it! This formula, ** a_n = 3 ⋅ 2^n-1**, allows us to calculate any term in this geometric sequence. For example, to find the 4th term, we would substitute n = 4 into the formula:

a₄ = 3 ⋅ 2^4-1 = 3 ⋅ 2³ = 3 ⋅ 8 = 24

So, the 4th term in the sequence is 24. We can verify this by listing out the first few terms: 3, 6, 12, 24...

Why Option B is Correct, and Others are Not

The question presents us with four options for the formula of the nth term. Let's examine each option and understand why only option B is correct:

• A. a_n = 3^n-1 ⋅ 2: This formula incorrectly uses the first term (3) as the base for the exponent and multiplies by the common ratio (2). This does not align with the general formula for geometric sequences.
• B. a_n = 3 ⋅ 2^n-1: This is the correct formula. It perfectly matches the result we obtained by substituting a₁ = 3 and r = 2 into the general formula. This formula correctly represents a geometric sequence with a first term of 3 and a common ratio of 2.
• C. a_n = 3 + 2(n-1): This formula represents an arithmetic sequence, not a geometric sequence. In an arithmetic sequence, we add a constant difference (instead of multiplying by a common ratio) to get to the next term. This formula would generate a sequence like 3, 5, 7, 9..., where the difference between terms is 2.
• D. a_n = 3(n-1) + 2: This formula also represents an arithmetic sequence, not a geometric sequence. It's simply a different way of expressing a linear relationship. This formula would generate a sequence like -1, 2, 5, 8..., which has a constant difference of 3.

Therefore, the only formula that accurately represents the nth term of a geometric sequence with a₁ = 3 and r = 2 is B. a_n = 3 ⋅ 2^n-1.

Common Mistakes and How to Avoid Them

When working with geometric sequences, it's easy to make a few common mistakes. Let's discuss these pitfalls and how to avoid them:

1. Confusing Geometric and Arithmetic Sequences: The biggest mistake is mixing up the formulas for geometric and arithmetic sequences. Remember, geometric sequences involve multiplication by a common ratio, while arithmetic sequences involve addition of a common difference. Always double-check whether the sequence is geometric or arithmetic before applying a formula.
2. Incorrectly Applying the Exponent: In the geometric sequence formula (a_n = a₁ ⋅ r^n-1), the exponent (n-1) only applies to the common ratio (r), not the first term (a₁). Make sure you calculate r^n-1 first and then multiply by a₁.
3. Forgetting the (n-1): The exponent is n-1, not just n. This is because we start with the first term, and it takes n-1 multiplications by the common ratio to reach the nth term. Forgetting to subtract 1 from n will lead to an incorrect result.
4. Misidentifying a₁ or r: Carefully identify the first term (a₁) and the common ratio (r) from the given sequence. A simple way to find the common ratio is to divide any term by its preceding term (e.g., a₂ / a₁).

By being mindful of these common mistakes, you can confidently and accurately work with geometric sequences.

Conclusion

In this article, we've explored the process of finding the formula for the nth term of a geometric sequence, focusing on the specific case where the first term (a₁) is 3 and the common ratio (r) is 2. We've learned the general formula (a_n = a₁ ⋅ r^n-1), applied it to our problem, and understood why option B (** a_n = 3 ⋅ 2^n-1**) is the correct answer. We've also discussed common mistakes to avoid when working with geometric sequences.

Understanding geometric sequences is crucial for various mathematical applications, from financial calculations (like compound interest) to scientific modeling (like population growth). By mastering the formula and its application, you'll gain a valuable tool for solving a wide range of problems.

To further expand your understanding of geometric sequences and related mathematical concepts, you can explore resources like Khan Academy's Geometric Sequences. This will provide you with additional explanations, examples, and practice problems to solidify your knowledge. Remember, practice is key to mastering any mathematical concept!