6th Term Of Sequence: -1, -3, -9, ... Solved!

by Felix Dubois 46 views

Hey guys! Let's dive into the world of sequences and figure out how to find the sixth term in the sequence: -1, -3, -9, ... This is a classic math problem that involves recognizing patterns and applying the right formulas. So, grab your thinking caps, and let's get started!

Understanding Sequences

Before we jump into solving 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, called terms, often follow a specific pattern or rule. Recognizing this pattern is key to finding missing terms, like the sixth one in our sequence.

Arithmetic vs. Geometric Sequences

There are two main types of sequences we usually encounter: arithmetic and geometric. It's crucial to distinguish between them because they have different formulas for finding terms.

  • Arithmetic Sequences: These sequences have a constant difference between consecutive terms. For example, 2, 4, 6, 8, ... is an arithmetic sequence where the common difference is 2.
  • Geometric Sequences: These sequences have a constant ratio between consecutive terms. Our sequence, -1, -3, -9, ..., is a geometric sequence. To see why, let's look at the ratio between terms.

Identifying the Pattern in Our Sequence

In our sequence, -1, -3, -9, ..., let's find the ratio between consecutive terms. To do this, we'll divide a term by its preceding term:

  • -3 / -1 = 3
  • -9 / -3 = 3

As you can see, the ratio between consecutive terms is a constant 3. This tells us that our sequence is indeed a geometric sequence, with a common ratio of 3. This is a critical step because knowing the type of sequence dictates how we approach finding the sixth term.

Finding the Sixth Term

Now that we've established that our sequence is geometric with a common ratio of 3, we can use the formula for the nth term of a geometric sequence. This formula is a powerful tool that allows us to find any term in the sequence without having to list out all the preceding terms.

The Formula for the nth Term of a Geometric Sequence

The formula is:

an = a1 * r^(n-1)

Where:

  • an is the nth term (the term we want to find)
  • a1 is the first term of the sequence
  • r is the common ratio
  • n is the term number we're looking for

Applying the Formula to Our Sequence

In our case:

  • a1 = -1 (the first term)
  • r = 3 (the common ratio)
  • n = 6 (we want to find the sixth term)

Plugging these values into the formula, we get:

a6 = -1 * 3^(6-1)
a6 = -1 * 3^5
a6 = -1 * 243
a6 = -243

So, the sixth term in the sequence is -243. Therefore, the correct answer is B. -243.

Step-by-Step Breakdown

To make it even clearer, let's break down the steps we took to find the sixth term:

  1. Identify the sequence type: We determined it was a geometric sequence.
  2. Find the common ratio: We calculated the common ratio (r) to be 3.
  3. Write down the formula: We recalled the formula for the nth term of a geometric sequence.
  4. Plug in the values: We substituted the values of a1, r, and n into the formula.
  5. Calculate: We performed the calculation to find the sixth term.

Understanding the Options and Why They Matter

Let's take a quick look at the options provided and why understanding the sequence helps us eliminate incorrect answers.

  • A. -729: This number is larger in magnitude than -243. If we hadn't used the formula and instead tried to continue the sequence manually, we might have made a mistake and arrived at this answer. However, by understanding the formula and applying it correctly, we avoid such errors.
  • B. -243: This is the correct answer, as we've already demonstrated.
  • C. -6561: This number is significantly larger in magnitude and is unlikely to be the sixth term if we correctly understand the growth pattern of the sequence.
  • D. -2187: This is another large number that could be a result of miscalculation or misunderstanding the sequence's pattern.

Common Mistakes to Avoid

When working with geometric sequences, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

  1. Confusing Arithmetic and Geometric Sequences: This is a major one! Using the wrong formula will lead to an incorrect answer. Always take the time to identify the type of sequence before proceeding.
  2. Miscalculating the Common Ratio: A wrong common ratio will throw off your entire calculation. Double-check your division to ensure accuracy.
  3. Incorrectly Applying the Exponent: Remember that the exponent in the formula is (n-1), not n. Forgetting to subtract 1 is a common mistake.
  4. Making Arithmetic Errors: Simple calculation mistakes can happen, especially with larger numbers. Take your time and double-check your work.

Practice Makes Perfect

Like any math skill, mastering sequences takes practice. The more you practice, the better you'll become at recognizing patterns and applying the formulas. Try working through different examples, varying the first term, common ratio, and term number you're trying to find. You can also challenge yourself by trying to find a general formula for a given sequence.

Real-World Applications of Sequences

You might be wondering,