Problem 59 - Calculus for Biology and Medicine

Question

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=3^{n} $$

Step-by-Step Solution

Answer

The first five terms are 1, 3, 9, 27, 81. The limit does not exist as \(a_n\to\infty\).

Step 1: Write the General Formula

We are given the general term of the sequence as \(a_n = 3^n\). This formula allows us to calculate any term in the sequence based on its position \(n\).

Step 2: Calculate the First Term

To find the first term \(a_0\), substitute \(n = 0\) into the formula: \(a_0 = 3^0 = 1\).

Step 3: Calculate the Second Term

Substitute \(n = 1\) into the formula to find \(a_1\): \(a_1 = 3^1 = 3\).

Step 4: Calculate the Third Term

Substitute \(n = 2\) into the formula to find \(a_2\): \(a_2 = 3^2 = 9\).

Step 5: Calculate the Fourth Term

Substitute \(n = 3\) into the formula to find \(a_3\): \(a_3 = 3^3 = 27\).

Step 6: Calculate the Fifth Term

Substitute \(n = 4\) into the formula to find \(a_4\): \(a_4 = 3^4 = 81\).

Step 7: Determine the Limit as \(n\to\infty\)

As \(n\) becomes very large, \(a_n = 3^n\) increases without bound since \(3^n\) grows exponentially. Thus, \(\lim_{n \to \infty} a_n = \infty\).

Key Concepts

Exponential Growth

Exponential growth is a powerful concept that describes a process where the quantity increases quickly over time based on a multiplication pattern. In our sequence exercise, we have the formula \( a_n = 3^n \). Here, we're witnessing exponential growth because each term in the sequence is three times the previous term. As \( n \) increases, \( 3^n \) becomes much larger quickly, illustrating how exponential functions expand:

• For \( n = 0 \): \( a_0 = 1 \)
• For \( n = 1 \): \( a_1 = 3 \)
• For \( n = 2 \): \( a_2 = 9 \)
• For \( n = 3 \): \( a_3 = 27 \)
• For \( n = 4 \): \( a_4 = 81 \)

By recognizing how exponential growth operates, it becomes easier to understand why the sequence rapidly grows as the term number increases.

Limit of a Sequence

The limit of a sequence examines the behavior of the sequence's terms as they approach a particular value when \( n \) (the position of the term) becomes very large. In this scenario, since \( a_n = 3^n \), as \( n \) approaches infinity, the sequence terms continue to grow vastly larger. Exponentially increasing values never settle down or approach a fixed number. Therefore, \( \lim_{n \to \infty} a_n = \infty \), indicating that there's no finite limit.Understanding limits is crucial because it shows whether the sequence is converging (approaching a definite number) or diverging (moving towards infinity). For any exponential sequence where the base is greater than one, like our example, the limit will typically point towards infinity.

General Term of a Sequence

The general term of a sequence is a formula that represents any term in the sequence based on its position \( n \). In our example, the general term is expressed as \( a_n = 3^n \).With this formula:

• You can plug in any value for \( n \) to find the respective sequence term.
• The pattern of the sequence becomes clear, since each term is derived from this expression.

Knowing the general term allows for quick calculations without manually generating a sequence from scratch each time. It's a tool to see how the sequence behaves, identify patterns, and predict long-term tendencies. This simplifies the process of understanding and analyzing the growth pattern and behavior of the sequence over time.