Problem 15 - Calculus for Biology and Medicine

Question

Determine the values of the sequence \(\left\\{a_{n}\right\\}\) for \(n=0,1,2, \ldots, 5\) $$ f(n)=\left(\frac{1}{3}\right)^{n} $$

Step-by-Step Solution

Answer

The sequence values are: \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}\).

Step 1: Define the Sequence Formula

The sequence is defined by the function \( f(n) = \left( \frac{1}{3} \right)^n \). This formula gives us the value of the sequence for each natural number \( n \).

Step 2: Calculate \( a_0 \)

Substitute \( n = 0 \) into the formula: \( a_0 = f(0) = \left( \frac{1}{3} \right)^0 = 1 \). The first term of the sequence is \( 1 \).

Step 3: Calculate \( a_1 \)

Substitute \( n = 1 \) into the formula: \( a_1 = f(1) = \left( \frac{1}{3} \right)^1 = \frac{1}{3} \). The second term of the sequence is \( \frac{1}{3} \).

Step 4: Calculate \( a_2 \)

Substitute \( n = 2 \) into the formula: \( a_2 = f(2) = \left( \frac{1}{3} \right)^2 = \frac{1}{9} \). The third term of the sequence is \( \frac{1}{9} \).

Step 5: Calculate \( a_3 \)

Substitute \( n = 3 \) into the formula: \( a_3 = f(3) = \left( \frac{1}{3} \right)^3 = \frac{1}{27} \). The fourth term of the sequence is \( \frac{1}{27} \).

Step 6: Calculate \( a_4 \)

Substitute \( n = 4 \) into the formula: \( a_4 = f(4) = \left( \frac{1}{3} \right)^4 = \frac{1}{81} \). The fifth term of the sequence is \( \frac{1}{81} \).

Step 7: Calculate \( a_5 \)

Substitute \( n = 5 \) into the formula: \( a_5 = f(5) = \left( \frac{1}{3} \right)^5 = \frac{1}{243} \). The sixth term of the sequence is \( \frac{1}{243} \).

Key Concepts

Understanding the Sequence Formula

At the heart of any geometric sequence is its sequence formula. This is a mathematical rule that defines how each term in the sequence is computed. In this case, the formula given is \( f(n) = \left( \frac{1}{3} \right)^n \). This formula helps us find the value of any term in the sequence by substituting different values of \( n \). The formula consists of two main components:

• A base, \( \frac{1}{3} \), which is consistent for all terms. This base is the common ratio of the sequence.
• An exponent, \( n \), which indicates which term we are calculating.

Each term in the sequence is thus a power of the base, increasing with each consecutive value of \( n \). Understanding this pattern allows us to efficiently compute any term in the sequence without needing prior terms, making it a powerful tool for quick calculations.

The Role of Exponents

Exponents play a crucial role in forming a geometric sequence. Essentially, an exponent tells us how many times to multiply the base by itself. In our sequence formula, \( \left( \frac{1}{3} \right)^n \), the exponent \( n \) increases as we calculate successive terms. This means:

• For \( n=0 \), the term is \( 1 \) because any number to the power of zero is \( 1 \).
• For \( n=1 \), the base stays as is, giving us \( \frac{1}{3} \).
• For \( n=2 \), the term becomes \( \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \).
• As \( n \) increases, the fraction becomes smaller because we are dividing by \( 3 \) multiple times.

This demonstrates how exponents can transform a base through repeated operations, allowing sequences to represent exponential decay, such as in this example.

Performing Term Calculation

To find the value of each term in the sequence, we apply the term calculation process using the sequence formula. This involves substituting values of \( n \) into the sequence equation \( f(n) = \left( \frac{1}{3} \right)^n \). Here's a breakdown of how to calculate the first few terms:

• Substituting \( n=0 \) gives \( a_0 = \left( \frac{1}{3} \right)^0 = 1 \).
• Substituting \( n=1 \) gives \( a_1 = \left( \frac{1}{3} \right)^1 = \frac{1}{3} \).
• Substituting \( n=2 \) gives \( a_2 = \left( \frac{1}{3} \right)^2 = \frac{1}{9} \).
• Continue this process for other values of \( n \).

This systematic approach ensures accuracy and helps to visualize how each successive term becomes a fraction of the last, making it easy to understand the diminishing nature of the sequence.