Problem 29 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \ldots\)

Step-by-Step Solution

Answer

The expression for \(a_{n}\) is \(\frac{1}{3^n}\).

Step 1: Analyze Given Values and Identify Pattern

The sequence given is \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \ldots\). Begin by observing the denominators: 1, 3, 9, 27, 81. Notice these are powers of 3. Specifically, they correspond to \(3^0, 3^1, 3^2, 3^3, 3^4\), indicating that each term of the sequence is \(\frac{1}{3^n}\), where \(n\) is the position in the sequence starting at 0.

Step 2: Generalize the Sequence Formula

Using the observation from Step 1, we can express the \(n\)-th term \(a_n\) of the sequence as \(\frac{1}{3^n}\). This accounts for all the observed terms: \(a_0=\frac{1}{3^0} = 1\), \(a_1=\frac{1}{3^1} = \frac{1}{3}\), \(a_2=\frac{1}{3^2} = \frac{1}{9}\), etc.

Step 3: Confirm the Expression with Examples

To ensure the correctness of the expression \(a_n = \frac{1}{3^n}\), plug in some values of \(n\): For \(n=0\), \(a_0 = \frac{1}{3^0} = 1\) matches the first term; for \(n=1\), \(a_1 = \frac{1}{3^1} = \frac{1}{3}\); for \(n=2\), \(a_2 = \frac{1}{3^2} = \frac{1}{9}\). This confirms our expression.

Key Concepts

Understanding Sequences

Sequences are lists of numbers arranged in a specific order. Each number in the sequence is termed a "term." The sequence provided: 1, \(\frac{1}{3}\), \(\frac{1}{9}\), \(\frac{1}{27}\), \(\frac{1}{81}\), ... is an example of a geometric sequence. In this type of sequence, each term is derived by multiplying the previous term by a constant value, known as the common ratio.
In a geometric sequence, the common ratio can be determined by dividing any term in the sequence by the preceding term. For this particular sequence, dividing \(\frac{1}{3}\) by 1 yields \(\frac{1}{3}\), and dividing \(\frac{1}{9}\) by \(\frac{1}{3}\) also yields \(\frac{1}{3}\). Thus, the common ratio is \(\frac{1}{3}\).
To find the general formula for a geometric sequence, you use the following expression:

• Every term can be expressed as \(a_n = a_0 \cdot r^n\)
• Where \(a_0\) is the first term and \(r\) is the common ratio

Substitute \(a_0 = 1\) and \(r = \frac{1}{3}\) into this equation to express terms in this sequence.

Identifying Patterns in Mathematics

Mathematical patterns refer to regularities or structures that can be identified in a set of numbers or figures. Patterns are crucial for predicting future terms and understanding deeper mathematical relationships.
In our sequence \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \ldots\), we observe a clear pattern in the powers of 3 that form the denominators. Recognizing these patterns helps us derive formulas and comprehend the inherent structure of sequences more effectively.
Recognizing patterns is also essential in various real-world applications, such as computer algorithms, data analysis, and even in predicting trends. By learning to spot patterns, you're building a foundation for solving complex mathematical problems and enhancing your analytical skills.

Exploring Exponential Functions

Exponential functions are powerful mathematical tools that describe situations where a quantity grows or decays at a constant relative rate. In simple terms, any function of the form \(f(x) = b^x\), where \(b\) is a constant, is considered exponential.
In our exercise, the powers of 3 leading to expressions like \(\frac{1}{3^n}\) illustrate exponential behavior. The term \(3^n\) denotes exponential growth of the denominator, whereas \(\frac{1}{3^n}\) signifies exponential decay of the fraction's overall value.
Understanding exponential functions and their features is essential, as these functions have vast applications in real life, such as in population dynamics, finance (compound interest), and natural phenomena like radioactive decay or growth.