Problem 25 - Calculus for Biology and Medicine

Question

In Problems 25-36, find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(0,1,2,3,4, \ldots\)

Step-by-Step Solution

Answer

The expression for \(a_n\) is \(a_n = n\).

Step 1: Analyze the Given Sequence

We are given a sequence where the values of the terms are: \(a_0 = 0\), \(a_1 = 1\), \(a_2 = 2\), and so on. Observe that the sequence appears to be an arithmetic sequence with an increment of 1 between each term.

Step 2: Identify the Pattern

In this sequence, the value of each term corresponds directly to its index, i.e., the term \(a_n\) equals \(n\). This suggests a straightforward relationship where the term value is exactly the same as the term index.

Step 3: Define the General Expression

Based on the pattern observed, the general expression for the sequence \(a_n = n\) can be articulated. This indicates that each term \(a_n\) is equal to its position \(n\) in the sequence.

Key Concepts

Sequence Analysis

When we talk about sequence analysis, it's all about understanding the list of numbers or terms provided to us. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

• In sequence analysis, our goal is to identify characteristics like the starting point, the common difference, and any distinct features of the sequence.
• In the original exercise given, the sequence starts at 0 and increments by 1 at each step.
• Importantly, each term is represented by 'n', the position it holds in the sequence.
• Therefore, the sequence is increasing by a common difference of 1.

Acknowledging this allows us to recognize the pattern that each subsequent term builds directly off the previous one. Hence, sequence analysis is the foundational step to understanding more advanced patterns and expressions.

Pattern Recognition

Pattern recognition in arithmetic sequences is about observing the regularities in the sequence that help us determine a formula.

• Once we've done the initial sequence analysis and noticed the regular increment of 1, pattern recognition comes into the picture.
• In our sequence, each term exactly matches its position index, indicating an extremely simple pattern.
• By seeing this, you can declare that each term increases by one as its index increases by one; thus, there isn’t just a unique relationship but a straightforward one-to-one correlation between the term and its index.

This insight from pattern recognition allows us to transition easily into defining a general expression.

General Expression

A general expression for an arithmetic sequence is a concise formula that describes any term in the sequence.

• In our example, the sequence pattern revealed that each term's value is exactly equal to its index.
• This simplifies our general expression for the sequence to be simply \(a_n = n\).
• This essentially means at any point, for any value of 'n', the term itself is precisely 'n'.

Having a general expression is powerful as it allows us to determine any term in the sequence without having to list or count out all the terms. It wraps the sequence into a single line that easily expresses the entirety of the sequence's behavior.