Finding A_11 In The Sequence A_n = 2^n - 5 Step By Step Solution

In this article, we will delve into the fascinating world of sequences and explore how to determine a specific term within a sequence defined by a formula. Specifically, we will focus on the sequence given by the formula a_n = 2^n - 5, and our goal is to find the 11th term, denoted as a_11. This involves substituting n with 11 in the given formula and performing the necessary calculations. Understanding how to find specific terms in a sequence is crucial in various mathematical applications, including calculus, discrete mathematics, and computer science. Let's embark on this mathematical journey and unravel the value of a_11.

Understanding Sequences and Their Terms

Before we dive into the specifics of finding a_11, it's essential to have a solid grasp of what sequences are and how their terms are defined. A sequence, in mathematical terms, is an ordered list of numbers, often following a specific pattern or rule. Each number in the sequence is referred to as a term, and the position of a term in the sequence is indicated by its subscript. For instance, in the sequence a_1, a_2, a_3, ..., a_n, a_1 represents the first term, a_2 represents the second term, and so on, with a_n representing the nth term.

Sequences can be defined in several ways, but one common method is by providing a formula that expresses the nth term as a function of n. This formula, often denoted as a_n, allows us to calculate any term in the sequence simply by substituting the desired value of n into the formula. This is particularly useful for finding terms that are far along in the sequence, as we don't need to calculate all the preceding terms. In our case, the sequence is defined by the formula a_n = 2^n - 5, which means that to find any term, we need to raise 2 to the power of the term's position and then subtract 5.

The beauty of using a formula to define a sequence lies in its ability to concisely represent an infinite number of terms. Instead of listing out each term individually, which would be impossible for an infinite sequence, the formula provides a compact and efficient way to describe the entire sequence. This makes it easier to analyze the sequence's properties, such as its growth rate, its limits (if any), and the relationships between its terms. Understanding the formula is the key to unlocking the secrets of the sequence and predicting its behavior.

The Formula for the Sequence: a_n = 2^n - 5

Now, let's take a closer look at the formula that defines our sequence: a_n = 2^n - 5. This formula is a simple yet powerful expression that dictates how each term in the sequence is generated. The formula tells us that to find the nth term, we need to perform two operations:

1. Raise 2 to the power of n: This means multiplying 2 by itself n times. For example, if n is 3, we calculate 2^3 = 2 * 2 * 2 = 8.
2. Subtract 5 from the result: After calculating 2^n, we subtract 5 from it to obtain the final value of the nth term. For instance, if 2^n is 8, then a_n = 8 - 5 = 3.

The exponential term, 2^n, plays a crucial role in the behavior of the sequence. As n increases, 2^n grows rapidly, leading to a sequence that increases exponentially. The subtraction of 5 simply shifts the entire sequence downwards but doesn't alter the exponential growth pattern. This understanding of the formula's components is vital for predicting how the sequence will behave as n gets larger.

To further illustrate how the formula works, let's calculate the first few terms of the sequence:

• a_1 = 2^1 - 5 = 2 - 5 = -3
• a_2 = 2^2 - 5 = 4 - 5 = -1
• a_3 = 2^3 - 5 = 8 - 5 = 3
• a_4 = 2^4 - 5 = 16 - 5 = 11

As we can see, the sequence starts with negative values and then quickly grows as n increases. This initial exploration of the sequence's terms provides valuable intuition for what to expect when we calculate a_11.

Finding the 11th Term: a_11

Our primary goal is to find the 11th term of the sequence, which is denoted as a_11. To do this, we simply substitute n with 11 in the formula a_n = 2^n - 5. This gives us:

a_11 = 2^11 - 5

Now, we need to calculate 2^11. This means multiplying 2 by itself 11 times:

2^11 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2048

So, 2 raised to the power of 11 is 2048. Now, we substitute this value back into our equation for a_11:

a_11 = 2048 - 5

Finally, we perform the subtraction:

a_11 = 2043

Therefore, the 11th term of the sequence defined by a_n = 2^n - 5 is 2043. This value highlights the rapid growth of the sequence due to the exponential term. The term a_11 is significantly larger than the initial terms we calculated earlier, demonstrating the power of exponential growth.

Importance of Finding Specific Terms in Sequences

Finding specific terms in a sequence, like we did with a_11, is not just a mathematical exercise; it has practical applications in various fields. Sequences are used to model a wide range of phenomena, from population growth and compound interest to the behavior of algorithms and the patterns in nature. Being able to determine a specific term in a sequence allows us to make predictions, analyze trends, and solve real-world problems.

For example, in finance, sequences can be used to model the growth of an investment over time. Each term in the sequence represents the value of the investment at a particular point in time, and the formula defining the sequence incorporates factors like the interest rate and the compounding frequency. By finding a specific term, we can estimate the value of the investment after a certain number of years. Similarly, in computer science, sequences are used to analyze the performance of algorithms. The number of operations an algorithm performs as the input size grows can be represented as a sequence, and finding a specific term can tell us how the algorithm will behave for a particular input size.

Furthermore, understanding how to find specific terms in sequences is crucial for more advanced mathematical concepts, such as series, limits, and calculus. Many of these concepts rely on the ability to analyze the behavior of sequences as the number of terms increases. Being able to calculate specific terms helps us develop an intuition for these behaviors and provides a foundation for more sophisticated mathematical analysis. In essence, the ability to find specific terms in sequences is a fundamental skill that underpins a wide range of mathematical and scientific applications.

Conclusion: a_11 = 2043

In conclusion, we have successfully found the 11th term of the sequence defined by a_n = 2^n - 5. By substituting n with 11 in the formula and performing the necessary calculations, we determined that a_11 = 2043. This exercise demonstrates the importance of understanding sequences and their formulas, as well as the ability to apply these concepts to find specific terms. The exponential nature of the sequence leads to rapid growth, as evidenced by the significant value of a_11 compared to the earlier terms. This skill of finding specific terms is a valuable tool in various mathematical and scientific contexts, allowing us to make predictions, analyze trends, and solve real-world problems. The world of sequences is vast and fascinating, and mastering the fundamentals, like finding specific terms, opens the door to a deeper understanding of mathematics and its applications.