Problem 4 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The absolute values of the sequence for \(n=0\) to \(5\) are: 0, \(\frac{1}{3}\), \(\frac{1}{2}\), \(\frac{3}{5}\), \(\frac{2}{3}\), \(\frac{5}{7}\).

Step 1: Understand the Sequence Formula

We are given the sequence \(a_n = \frac{n}{n+2}\). Our task is to find the absolute values of this sequence for \(n = 0, 1, 2, 3, 4, 5\), i.e., \(\left|a_n\right|\).

Step 2: Calculate Values for Each n

We will calculate \(a_n\) for each \(n\) from 0 to 5 and then take the absolute value.- For \(n = 0\), \(a_0 = \frac{0}{0+2} = 0\) \(\left|a_0\right| = 0\).- For \(n = 1\), \(a_1 = \frac{1}{1+2} = \frac{1}{3}\) \(\left|a_1\right| = \frac{1}{3}\).- For \(n = 2\), \(a_2 = \frac{2}{2+2} = \frac{2}{4} = \frac{1}{2}\) \(\left|a_2\right| = \frac{1}{2}\).- For \(n = 3\), \(a_3 = \frac{3}{3+2} = \frac{3}{5}\) \(\left|a_3\right| = \frac{3}{5}\).- For \(n = 4\), \(a_4 = \frac{4}{4+2} = \frac{4}{6} = \frac{2}{3}\) \(\left|a_4\right| = \frac{2}{3}\).- For \(n = 5\), \(a_5 = \frac{5}{5+2} = \frac{5}{7}\) \(\left|a_5\right| = \frac{5}{7}\).

Step 3: List Absolute Values

The sequence of absolute values \(\left|a_n\right|\) for \(n=0,1,2,3,4,5\) is:- \(n = 0\): \(0\)- \(n = 1\): \(\frac{1}{3}\)- \(n = 2\): \(\frac{1}{2}\)- \(n = 3\): \(\frac{3}{5}\)- \(n = 4\): \(\frac{2}{3}\)- \(n = 5\): \(\frac{5}{7}\)

Key Concepts

Understanding Absolute Value in Sequences

The absolute value of a number is essentially its distance from zero on the number line, without considering its direction. Whether the original number is positive or negative, the absolute value is always non-negative. To calculate the absolute value of any real number \(x\), you use the notation \(|x|\), pronounced as the "absolute value of \(x\)." For positive numbers, or zero, the absolute value is simply the same number. For negative numbers, you need to multiply by \(-1\) to make it positive, which gives us the rule:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

In this sequence problem, since all the expressions \(a_n = \frac{n}{n+2}\) for \(n = 0, 1, 2, 3, 4, 5\) result in fractions between 0 and 1, the absolute value does not alter the values. Thus \(\left|a_n\right| = \frac{n}{n+2}\) for the range we are interested in.

Exploring Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are similar to regular fractions, but instead of whole numbers, they include variables. For example, in the expression \(a_n = \frac{n}{n+2}\), \(n\) is the variable that makes it a rational expression.
Simplifying rational expressions often involves factoring where possible, canceling out common factors, or understanding their behavior at various points. In our case, since \(n\) is a non-negative integer, the expression behaves well and is straightforward.

• Numerator: \(n\)
• Denominator: \(n + 2\)

This expression simplifies for each specific value of \(n\). Calculating it for various values of \(n\) gives useful insight into how the sequence behaves. The denominator \(n + 2\) ensures that as \(n\) increases, the value of the rational expression \(\frac{n}{n+2}\) approaches 1, but never exceeds it.

The Step-by-Step Solution Approach

A step-by-step solution is a methodical way to solve a problem by breaking it down into smaller, manageable parts. This approach ensures that each part of the problem is correctly addressed, which minimizes errors and builds a clear understanding of the entire process. In our sequence analysis exercise, the step-by-step approach involved:

• Identifying the Problem: Understanding what the sequence formula is and deciding the values for which the sequence needs to be analyzed (here, from \(n = 0\) to \(n = 5\)).
• Substitution: For each \(n\), substitute into the sequence formula, \(a_n = \frac{n}{n+2}\) to find specific numbers.
• Calculating Absolute Values: Determine the non-negative results of these values to get \(\left|a_n\right|\).
• Compiling Results: Listing the absolute values for each \(n\) calculated. This yields the final sequence desired.

By following each of these steps systematically, we ensure accuracy and a thorough understanding of not just the solution, but also the process employed to arrive at that solution.