Problem 10 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The values of the sequence \(|a_n|\) for \(n = 0,1,2,\ldots,5\) are 2, 0, 2, 0, 2, 0 respectively.

Step 1: Define the Sequence

The sequence given is \( a_n = (-1)^n + 1 \). You are required to find the values of \( |a_n| \) for \( n = 0, 1, 2, \ldots, 5 \).

Step 2: Compute for n = 0

Substitute \( n = 0 \) into the sequence formula: \[ a_0 = (-1)^0 + 1 = 1 + 1 = 2 \]Since absolute values are non-negative, \( |a_0| = 2 \).

Step 3: Compute for n = 1

Substitute \( n = 1 \) into the sequence formula:\[ a_1 = (-1)^1 + 1 = -1 + 1 = 0 \]Thus, \( |a_1| = 0 \).

Step 4: Compute for n = 2

Substitute \( n = 2 \) into the sequence formula:\[ a_2 = (-1)^2 + 1 = 1 + 1 = 2 \]Therefore, \( |a_2| = 2 \).

Step 5: Compute for n = 3

Substitute \( n = 3 \) into the sequence formula:\[ a_3 = (-1)^3 + 1 = -1 + 1 = 0 \]This gives \( |a_3| = 0 \).

Step 6: Compute for n = 4

Substitute \( n = 4 \) into the sequence formula:\[ a_4 = (-1)^4 + 1 = 1 + 1 = 2 \]Thus, \( |a_4| = 2 \).

Step 7: Compute for n = 5

Substitute \( n = 5 \) into the sequence formula:\[ a_5 = (-1)^5 + 1 = -1 + 1 = 0 \]Therefore, \( |a_5| = 0 \).

Key Concepts

Absolute Values

Absolute values are a fundamental concept in mathematics. They provide the non-negative magnitude of a number or expression, regardless of its sign. Think of absolute values as a way to measure distance from zero without worrying about direction.

• For any real number \(x\), the absolute value is represented by \(|x|\).
• It is always non-negative, meaning \(|x| \geq 0\).
• If \(x\) is positive or zero, \(|x| = x\), and if \(x\) is negative, \(|x| = -x\).

In sequence analysis, the absolute value is often used to simplify calculations or to ensure that values are presented positively, which can be particularly helpful in identifying behavior and trends in a sequence. In our exercise, the absolute values of the sequence \(a_n = (-1)^n + 1\) were calculated, showing how negative values were turned into zero during computations.

Sequence Analysis

Sequence analysis involves identifying patterns and properties within a sequence of numbers. Each sequence has a defined rule or formula that tells us how the next number is generated. In mathematics, sequences are a list of numbers where each term has a specific position or index.

• The general form of a sequence can be written as \( \{a_n\} \).
• The position \(n\) is a non-negative integer that indicates the term's place in the sequence, starting from \(n=0\) or \(n=1\).
• A single term in the sequence is often denoted \(a_n\), where \(n\) identifies its order in the series.

Analyzing the sequence given \(a_n = (-1)^n + 1\), we notice a pattern where each term alternates between 2 and 0. This property allows us to identify and validate the specific behavior for a range of values \(n=0\) to \(n=5\), offering insights into the structure and regularity of the sequence.

Alternating Series

An alternating series is a sequence in which the terms change sign. This means the elements of the series alternate between positive and negative, which is highlighted by a factor of \((-1)^n\) within its formula.

• In our exercise, \(a_n = (-1)^n + 1\) is defined as an alternating series.
• For even \(n\), \((-1)^n\) is positive, leading to positive terms.
• For odd \(n\), \((-1)^n\) is negative, causing those terms to adjust accordingly (here, becoming zero).

Understanding alternating series is crucial since they often appear in mathematical contexts where oscillation or fluctuation is relevant. In our example, it underscores how sequence values oscillate, going from one value to another based on whether \(n\) is even or odd. This regular switching provides an excellent introduction to more complex series and functions with similar behavior.