3rd Term Of Sequence: A(n) = (3/2)(-2)^(n-1)

Hey guys! Today, we're diving into the exciting world of sequences and series. Specifically, we're going to tackle a problem where we need to find the 3rd term of a sequence defined by the formula $a(n) = \frac{3}{2}(-2)^{n-1}$. Don't worry if that looks a bit intimidating – we'll break it down step by step so it's super easy to understand. So, grab your thinking caps, and let's get started!

Understanding Sequences

Before we jump into the problem, let's make sure we're all on the same page about what a sequence actually is. In mathematics, a sequence is simply an ordered list of numbers. These numbers are called terms, and they often follow a specific pattern or rule. Think of it like a line of dominoes – each domino is a term, and the way they're arranged follows a certain rule. For example, we might have a sequence like 2, 4, 6, 8, 10, where each term is obtained by adding 2 to the previous term. This is a pretty straightforward sequence, but they can get much more complex!

Sequences can be finite, meaning they have a specific number of terms (like our example above), or they can be infinite, meaning they go on forever. We often use a formula to describe the nth term of a sequence, which allows us to find any term in the sequence without having to list them all out. This is where our equation $a(n) = \frac{3}{2}(-2)^{n-1}$ comes into play. This formula tells us exactly how to calculate any term in this particular sequence, just by plugging in the term number (n).

Understanding the formula is key. Let’s break it down. The $a(n)$ part simply means "the nth term of the sequence." The ${\frac{3}{2}}$ is a constant multiplier. The $(-2)^{n-1}$ part is where the magic happens – it involves raising -2 to the power of (n - 1), which means the term number minus 1. This is what creates the pattern in the sequence. By understanding each part of the formula, we’re setting ourselves up for success in finding the 3rd term.

Identifying the Formula and the Goal

Now, let's zoom in on the specific problem we're tackling. We're given the formula for the sequence: $a(n) = \frac{3}{2}(-2)^{n-1}$. This is our explicit formula, which means it directly tells us how to calculate any term in the sequence. It's like having a recipe – you know exactly what ingredients (or operations) you need to get the desired result.

The question asks us to find the 3rd term in the sequence. In mathematical terms, this means we need to find the value of $a(3)$. The “3” here represents the term number, so we’re looking for the third number in the sequence. This is our goal: to determine the value of $a(3)$ using the given formula.

To recap, we have a formula that defines the sequence, and we have a specific term we need to find. This is a classic problem in sequences and series, and it highlights the power of using formulas to describe patterns. By understanding what the formula represents and what the question is asking, we can confidently move forward to the calculation step. It's like knowing where you're going before you start driving – you're much more likely to reach your destination!

Step-by-Step Calculation

Alright, let's get down to business and calculate the 3rd term! Remember, our formula is $a(n) = \frac{3}{2}(-2)^{n-1}$, and we want to find $a(3)$. This means we're going to substitute n with 3 in the formula. Think of it as replacing a placeholder with a specific value.

Here's the first step: replace n with 3:

$a(3) = \frac{3}{2}(-2)^{3-1}$

Now, let's simplify the exponent. 3 - 1 equals 2, so we have:

$a(3) = \frac{3}{2}(-2)^{2}$

Next, we need to evaluate $(-2)^2$. Remember, this means -2 multiplied by itself: (-2) * (-2) = 4. A negative number multiplied by a negative number gives a positive result. So, our equation becomes:

$a(3) = \frac{3}{2}(4)$

Finally, we multiply ${\frac{3}{2}}$ by 4. We can think of 4 as ${\frac{4}{1}}$, so we're multiplying two fractions: ${\frac{3}{2} * \frac{4}{1}}$. To multiply fractions, we multiply the numerators (top numbers) and the denominators (bottom numbers):

$a(3) = \frac{3 * 4}{2 * 1} = \frac{12}{2}$

And now, we simplify the fraction ${\frac{12}{2}}$. 12 divided by 2 is 6. So, we have:

$a(3) = 6$

There you have it! The 3rd term in the sequence is 6. We've successfully calculated the value by carefully substituting the term number into the formula and following the order of operations. Each step is like a piece of the puzzle, and when we put them together, we get the final answer.

Verification and Conclusion

Okay, we've arrived at our answer: $a(3) = 6$. But before we celebrate, it's always a good idea to verify our result. This is like double-checking your work in any task – it helps catch any potential errors and ensures we're confident in our solution.

One way to verify is to think about the pattern of the sequence. While we only calculated the 3rd term, we could also calculate the first few terms to see if our answer makes sense in the overall sequence. Let's calculate the first few terms:

• For n = 1: $a(1) = \frac{3}{2}(-2)^{1-1} = \frac{3}{2}(-2)^0 = \frac{3}{2}(1) = \frac{3}{2}$
• For n = 2: $a(2) = \frac{3}{2}(-2)^{2-1} = \frac{3}{2}(-2)^1 = \frac{3}{2}(-2) = -3$
• For n = 3: $a(3) = \frac{3}{2}(-2)^{3-1} = \frac{3}{2}(-2)^2 = \frac{3}{2}(4) = 6$ (This matches our calculated answer!)
• For n = 4: $a(4) = \frac{3}{2}(-2)^{4-1} = \frac{3}{2}(-2)^3 = \frac{3}{2}(-8) = -12$

So, the sequence starts like this: ${\frac{3}{2}}$, -3, 6, -12, ...

We can see that our calculated 3rd term, 6, fits nicely into this pattern. This verification step gives us extra confidence that our solution is correct. It's like having a map to confirm you've reached your destination. Therefore, we can confidently conclude that the 3rd term in the sequence defined by $a(n) = \frac{3}{2}(-2)^{n-1}$ is indeed 6. You nailed it! Understanding sequences and how to use formulas to find specific terms is a fundamental skill in mathematics. Keep practicing, and you'll become a sequence superstar in no time!