Problem 26 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ 0,2,4,6,8, \ldots $$

Step-by-Step Solution

Answer

The expression for \( a_n \) is \( 2n \).

Step 1: Identify the pattern

Examine the given sequence: 0, 2, 4, 6, 8, .... You can see that each term is increasing by a constant difference of 2. This indicates an arithmetic sequence.

Step 2: Understand the sequence structure

In an arithmetic sequence, the general term can be expressed as: \( a_n = a_0 + nd \), where \( a_0 \) is the first term and \( d \) is the common difference. Here, \( a_0 = 0 \) and \( d = 2 \).

Step 3: Plug values into the formula

Substitute the values from Step 2 into the arithmetic sequence formula. This results in: \( a_n = 0 + 2n = 2n \).

Step 4: Verify the expression

Check the expression \( a_n = 2n \) by substituting different values for \( n \). For example, if \( n=0 \), then \( a_0=0 \); if \( n=1 \), then \( a_1=2 \); if \( n=2 \), then \( a_2=4 \); and so on, which matches the given sequence.

Key Concepts

Sequence Pattern

In mathematics, a sequence is a set of numbers arranged in a specific order. For this exercise, the sequence provided is 0, 2, 4, 6, 8, and it continues indefinitely. Recognizing the pattern in a sequence is key to understanding how to work with it.

• Here, the pattern can be observed in the way each number in the sequence increases as compared to the previous number.
• Every number after the initial term increases by 2.
• This constant increment is what identifies the sequence as an arithmetic sequence.

A sequence pattern is essentially a roadmap for identifying how sequences progress, which is essential for predicting future numbers in arithmetic sequences or finding specific terms.

General Term Formula

The general term formula of an arithmetic sequence helps you find any term in the sequence without having to write out all the terms. It’s like having a shortcut to work out how sequences are put together.
To derive the general term formula, we use:

• Formula: \(a_n = a_0 + nd \)
• \(a_0\): the first term of the sequence
• \(n\): the term position you're looking to find
• \(d\): the common difference between terms

In our specific case,

• The first term, \(a_0\), is 0.
• The common difference, \(d\), here is 2.

Therefore, substituting these values into the formula gives us \(a_n = 0 + 2n = 2n\). This means that if you know the position of the term in the sequence, you can find its value just by multiplying the position number by 2.

Common Difference

The common difference in an arithmetic sequence is a value that represents how much each term in the sequence increases as you move from one term to the next.

• In an arithmetic sequence, this difference remains constant.
• Recognizing this common difference allows you to understand the sequence structure and to establish the pattern.

For the sequence 0, 2, 4, 6, 8, the common difference can easily be found by subtracting any term from the preceding term, such as:

• \(2 - 0 = 2\)
• \(4 - 2 = 2\)
• \(6 - 4 = 2\)

As seen, the difference is clearly 2 across all these terms. This is the basis for our general term formula, where each subsequent term is found by adding this consistent common difference to the previous term. Understanding the common difference is crucial to constructing accurate arithmetic sequences and performing calculations related to them.