Problem 2 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The values are 0, 3, 12, 27, 48, 75.

Step 1: Understand the Sequence Formula

The sequence given is described by the formula \( a_n = 3n^2 \). This means that for any value of \( n \), the corresponding value of \( a_n \) is obtained by substituting \( n \) into the formula and calculating \( 3n^2 \).

Step 2: Calculate \( |a_0| \)

For \( n = 0 \), substitute into the formula: \( a_0 = 3(0)^2 = 0 \). Therefore, \( |a_0| = |0| = 0 \).

Step 3: Calculate \( |a_1| \)

For \( n = 1 \), substitute into the formula: \( a_1 = 3(1)^2 = 3 \). Therefore, \( |a_1| = |3| = 3 \).

Step 4: Calculate \( |a_2| \)

For \( n = 2 \), substitute into the formula: \( a_2 = 3(2)^2 = 12 \). Therefore, \( |a_2| = |12| = 12 \).

Step 5: Calculate \( |a_3| \)

For \( n = 3 \), substitute into the formula: \( a_3 = 3(3)^2 = 27 \). Therefore, \( |a_3| = |27| = 27 \).

Step 6: Calculate \( |a_4| \)

For \( n = 4 \), substitute into the formula: \( a_4 = 3(4)^2 = 48 \). Therefore, \( |a_4| = |48| = 48 \).

Step 7: Calculate \( |a_5| \)

For \( n = 5 \), substitute into the formula: \( a_5 = 3(5)^2 = 75 \). Therefore, \( |a_5| = |75| = 75 \).

Key Concepts

Understanding Absolute Value

When dealing with sequences, particularly ones involving expressions, we often encounter the concept of absolute value. The absolute value of a number, denoted by vertical bars like this: \( |x| \), represents its distance from zero on the number line. It is always a non-negative value, regardless of whether the original number was positive or negative. For example:

• \( |3| = 3 \)
• \( |-3| = 3 \)
• \( |0| = 0 \)

In our exercise, we calculated the absolute value of a sequence \( |a_n| \), where \( a_n \) could never be negative because \( a_n = 3n^2 \) is always non-negative for any integer \( n \). Thus, in this particular problem, the absolute value mirrors the original sequence values.

Identifying a Quadratic Sequence

A quadratic sequence is a collection of numbers in which the general term is a quadratic function of its position in the sequence. For example, the sequence given in the exercise is defined by \( a_n = 3n^2 \). Here, \( n \) represents the position in the sequence, and \( 3n^2 \) is a quadratic expression of \( n \).The defining property of a quadratic sequence is the presence of the term \( n^2 \), making the sequence grow in a non-linear fashion. Quadratic sequences, like the one in our exercise,

• have a second difference that is constant, and
• typically increase more rapidly than linear sequences as \( n \) grows.

Understanding this characteristic helps in predicting sequence behavior and in identifying potential patterns within the sequence for further analysis.

Sequence Calculation Fundamentals

Calculating sequence values typically involves substituting values of \( n \) into a given formula to determine each term in the sequence. In our example, we have the formula \( a_n = 3n^2 \). To find the terms:1. Substitute \( n \) into the sequence formula.2. Evaluate the expression by performing the necessary arithmetic operations.Let's break down the calculation process with an example: To find \( |a_3| \):

• First calculate \( a_3 = 3 \times (3)^2 = 27 \)
• Then find the absolute value \(|a_3| = |27| = 27 \)

This process allows us to compute sequence values systematically. Understanding these fundamentals is crucial not just for solving specific problems, but also for grasping more complex sequence-related concepts in mathematics.