Problem 9 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The sequence is \(a_0 = 0\), \(a_1 = -1\), \(a_2 = 2\), \(a_3 = -3\), \(a_4 = 4\), \(a_5 = -5\).

Step 1: Understanding the Sequence Formula

The sequence is defined as \(a_n = (-1)^n n\). This means the value of each term depends on \((-1)^n\) and \(n\). The exponent \(n\) influences the sign of the term: if \(n\) is even, \((-1)^n = 1\), and if \(n\) is odd, \((-1)^n = -1\).

Step 2: Calculating \(a_0\)

For \(n = 0\), the formula gives \(a_0 = (-1)^0 \cdot 0 = 1 \cdot 0 = 0\).

Step 3: Calculating \(a_1\)

For \(n = 1\), the formula gives \(a_1 = (-1)^1 \cdot 1 = -1 \cdot 1 = -1\).

Step 4: Calculating \(a_2\)

For \(n = 2\), the formula gives \(a_2 = (-1)^2 \cdot 2 = 1 \cdot 2 = 2\).

Step 5: Calculating \(a_3\)

For \(n = 3\), the formula gives \(a_3 = (-1)^3 \cdot 3 = -1 \cdot 3 = -3\).

Step 6: Calculating \(a_4\)

For \(n = 4\), the formula gives \(a_4 = (-1)^4 \cdot 4 = 1 \cdot 4 = 4\).

Step 7: Calculating \(a_5\)

For \(n = 5\), the formula gives \(a_5 = (-1)^5 \cdot 5 = -1 \cdot 5 = -5\).

Key Concepts

Sequence Formula

A sequence formula allows you to find the value of each term in a sequence based on its position, often denoted as \(n\). In this case, the sequence is represented by \(a_n = (-1)^n n\). Each term is calculated using two components:

• The multiplier: \((-1)^n\) - This part decides the sign of each term. When \(n\) is even, \((-1)^n\) equals \(1\), making the term positive. Conversely, when \(n\) is odd, \((-1)^n\) equals \(-1\), turning the term negative.
• The term number: \(n\) - This is the simple multiplication factor indicating the sequence's natural number position. It also interacts with the multiplier to determine the term's final value.

By plugging integer values into \(n\), you calculate each term's value in the sequence. For example, at \(n = 2\), the calculation \((1\cdot2)\) results in \(2\), demonstrating the method's simplicity and effectiveness.

Even and Odd Numbers

Understanding even and odd numbers is crucial when dealing with sequences like \(a_n = (-1)^n n\). Even and odd numbers exhibit distinct patterns that affect calculations:

• Even numbers: Divisible by 2. Examples include 0, 2, 4, etc. For even \(n\), the expression \((-1)^n\) simplifies to \(1\), so the sequence term \(a_n = n\).
• Odd numbers: Not divisible by 2. Examples include 1, 3, 5, etc. For odd \(n\), the expression \((-1)^n\) simplifies to \(-1\), so the sequence term \(a_n = -n\).

This alternation between positive and negative values based on whether \(n\) is even or odd is the backbone of this sequence. It's easy to predict the outcome of each sequence term once you've determined if \(n\) is even or odd.

Alternating Sequences

An alternating sequence is one that changes sign with each successive term. In the sequence given by \(a_n = (-1)^n n\), the sign-change pattern is driven by the term \((-1)^n\). Here's a breakdown:

• For even \(n\), the term remains positive or zero because \((-1)^n\) results in 1.
• For odd \(n\), the term shifts to negative since \((-1)^n\) results in -1.

This results in each sequence term switching from positive to negative, which is referred to as alternating. These types of sequences are important in calculus as they model phenomena that's periodic or cyclical in nature. Understanding how alternation works helps in predicting the sequence's behavior and applying it in real-world or theoretical contexts.