Geometric Sequence: Is It Increasing Or Decreasing?

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Hey guys! Let's dive into the fascinating world of geometric sequences and figure out whether a particular sequence is increasing or decreasing. We'll break down a specific problem step by step, so you can confidently tackle similar questions. So, buckle up, and let’s get started!

Understanding Geometric Sequences

Before we jump into the problem, let's quickly recap what a geometric sequence actually is. A geometric sequence is a sequence of numbers where each term is multiplied by a constant value to get the next term. This constant value is called the common ratio, often denoted by 'q'. Imagine you start with a number, say 2, and keep multiplying it by 3. You'd get 2, 6, 18, 54, and so on. That’s a geometric sequence with a common ratio of 3. The general form of a geometric sequence is a_n = a_1 * q^(n-1), where a_n is the nth term, a_1 is the first term, and q is the common ratio. Understanding this basic formula is crucial for solving problems related to geometric sequences. We need to know how each element relates to the others and how the sequence behaves based on the common ratio. A solid grasp of these fundamentals will make determining whether a sequence is increasing or decreasing much easier. Knowing the formula helps us predict the trend of the sequence as 'n' increases. For example, if 'q' is greater than 1, the terms tend to grow larger, while if 'q' is between 0 and 1, the terms tend to shrink. This concept is essential for analyzing the behavior of different geometric sequences.

The Problem: Analyzing a_n = -3 * 2^(n-1)

Now, let's look at the problem at hand. We're given a geometric sequence (a_n) defined by the formula a_n = -3 * 2^(n-1). This formula tells us exactly how to find any term in the sequence. For instance, to find the first term (a_1), we substitute n = 1 into the formula. To find the second term (a_2), we substitute n = 2, and so on. The key question we need to answer is: Is this sequence increasing, decreasing, or neither? To answer this, we need to consider how the terms change as 'n' gets larger. Think about it – what happens when you keep multiplying by 2 in this context? Remember the -3 in front? That's a crucial part of the puzzle. To figure out the sequence's behavior, we need to analyze the common ratio and the initial term. The common ratio, in this case, is 2, which means each term is multiplied by 2 to get the next term. However, the initial term is -3, which is negative. This combination of a negative initial term and a common ratio greater than 1 will give us a particular pattern that we need to identify. Analyzing the effect of multiplying a negative number by an increasing power of 2 is essential to understanding the overall trend of the sequence. We need to carefully consider how these two factors interact to determine the sequence's behavior.

Finding the Common Ratio (q)

The common ratio, 'q', is the heart of a geometric sequence. It's the factor by which we multiply each term to get the next one. In our case, the formula a_n = -3 * 2^(n-1) pretty much gives it away. We can see that each term is being multiplied by a power of 2. The base of this exponent, which is 2, is our common ratio. So, q = 2. But, let’s say you weren't immediately sure. There's another way to find 'q'. Remember, in a geometric sequence, the ratio between any two consecutive terms is constant and equal to 'q'. That means a_(n+1) / a_n = q. So, we could calculate a few terms and divide them to double-check. For example, let's find a_1 and a_2. a_1 = -3 * 2^(1-1) = -3 * 2^0 = -3. a_2 = -3 * 2^(2-1) = -3 * 2^1 = -6. Now, if we divide a_2 by a_1, we get -6 / -3 = 2, which confirms our common ratio. This method is particularly useful if the common ratio isn't immediately obvious from the general term formula. It provides a concrete way to verify the common ratio and ensure we're on the right track. Knowing the common ratio helps us predict the direction of the sequence, whether it will increase, decrease, or alternate.

Determining the Sequence's Behavior

Now comes the crucial step: figuring out if the sequence is increasing, decreasing, or neither. We know the common ratio q = 2, which is greater than 1. This tells us that the absolute value of the terms will increase as 'n' increases. But, we also have that pesky -3 at the beginning. Let's calculate the first few terms to see what's going on:

  • a_1 = -3 * 2^(1-1) = -3
  • a_2 = -3 * 2^(2-1) = -6
  • a_3 = -3 * 2^(3-1) = -12
  • a_4 = -3 * 2^(4-1) = -24

Notice anything? The terms are getting more and more negative. Remember, on the number line, numbers get smaller as they move further to the left (more negative). So, -24 is smaller than -12, which is smaller than -6, and so on. Therefore, the sequence is decreasing. It's a common mistake to think that because the common ratio is greater than 1, the sequence must be increasing. However, the negative initial term flips everything around. This highlights the importance of considering both the common ratio and the initial term when analyzing the behavior of a geometric sequence. It's also crucial to remember that