Question

In Exercises \(1-8,\) write the first five terms of the sequence. \(a_{n}=2^{n}\)

Short answer

The first five terms of the sequence \(a_{n}=2^{n}\) are 2, 4, 8, 16, and 32.

Step-by-step solution

Find the first term

Replace \(n\) with 1 in the formula \(a_{n}=2^{n}\). Therefore, \(a_{1}=2^{1}=2\), which shows that the first term of the sequence is 2.

Find the second term

Replace \(n\) with 2 in the formula \(a_{n}=2^{n}\). Therefore, \(a_{2}=2^{2}=4\), which shows that the second term of the sequence is 4.

Find the third term

Replace \(n\) with 3 in the formula \(a_{n}=2^{n}\). Therefore, \(a_{3}=2^{3}=8\), which shows that the third term of the sequence is 8.

Find the fourth term

Replace \(n\) with 4 in the formula \(a_{n}=2^{n}\). Therefore, \(a_{4}=2^{4}=16\), which shows that the fourth term of the sequence is 16.

Find the fifth term

Replace \(n\) with 5 in the formula \(a_{n}=2^{n}\). Therefore, \(a_{5}=2^{5}=32\), which shows that the fifth term of the sequence is 32.

Key concepts

Terms of a Sequence

When we speak of the terms of a sequence, we refer to the individual elements that make up the sequence. For example, in the sequence given by the exponential function a_{n}=2^{n}, the terms are simply the outputs you get when you plug in consecutive integers for 'n'. Just like in the solved problem, the first term, labeled as a_{1}, is obtained by substituting n with 1, yielding 2^{1}=2. This process is continued to find subsequent terms.
To understand this better, let's consider a sequence as a kind of list, where each item in the list is generated by following a specific rule - in this case, the rule is 2^{n}. By following this rule, the terms generated create a pattern which can be predictable once we grasp the underlying sequence formula. Recognizing these patterns is essential for understanding sequences in mathematics and can be applied to various scenarios, such as calculating interest, analyzing sound waves, and even determining population growth.

The Pattern of the Sequence

In our specific example, we observe an exponential growth pattern, where each term is twice the value of the preceding term (since 2^n doubles as n increases by 1). This exponential behavior is quite common in the real-world phenomena, like radioactive decay and bacterial growth.
Recognizing and understanding the behavior of sequences through their terms is not only fundamental in algebra but is also an essential component of predictive analytics and computer algorithms.

Exponential Functions

The term exponential function might seem daunting at first, but it describes a very simple yet powerful mathematical concept. Exponential functions involve constant bases raised to the power of a variable exponent and are expressed in general form as f(x) = b^x, where b is a positive constant. Our example sequence a_{n}=2^{n} is one such exponential function where the base is 2.
One key characteristic of exponential functions is their rapid growth or decay, and they're often used in fields dealing with growth patterns, such as biology, finance, and physics. It's fascinating to see how a simple formula can model complex phenomena like the spread of viruses, compound interest in a bank account, or the cooling of a hot coffee.

Real-Life Applications

The principles governing exponential functions extend well into everyday life. For instance, understanding the exponential growth of savings and investments can lead to more informed financial decisions. Likewise, biologists use exponential decay functions to model population decreases in ecosystems. Mastering the exponential function is a valuable skill, as it enables us to model a myriad of natural and societal processes with great accuracy and predict future outcomes based on present data.

Mathematical Induction

Moving to a slightly different but deeply connected concept, mathematical induction is a powerful tool used to prove a wide range of mathematical statements. Essentially, it involves two crucial steps. First, you show that a statement holds for an initial case, usually n=1. Then, assuming the statement is true for n=k, you prove it for n=k+1. If both steps are successful, you've used induction to show that the statement is true for all positive integers.
In relation to sequences, induction can be used to prove properties about them. For example, we might want to prove that all terms in a certain sequence are positive, or that each term is greater than the term before it. For an exponential sequence like a_{n}=2^{n}, mathematical induction can neatly demonstrate why each subsequent term is indeed twice the previous term, thereby guaranteeing the pattern of the sequence.

Importance in Mathematics

Mathematical induction plays a crucial role in mathematics not just for sequences but across a variety of topics. It's used to prove theorems in number theory, the correctness of algorithms in computer science, and properties of geometric objects. The beauty of induction lies in its approach; by starting from a base case and then establishing a rule to build upon, it mirrors the way in which we intuitively learn from specific instances to general rules.