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 sequence provided: \(0, 2, 4, 6, 8, \ldots\). Note that each term increases by 2. This suggests an arithmetic sequence.

Step 2: Determine the common difference

In this sequence, each term increases by 2. Therefore, the common difference, \(d\), is 2.

Step 3: Identify the first term

The first term of the sequence, \(a_0\), is 0.

Step 4: Determine the formula of the sequence

For an arithmetic sequence, the formula for the \(n\)-th term \(a_n\) is given by \(a_n = a_0 + nd\). Substitute \(a_0 = 0\) and \(d = 2\) to get \(a_n = 0 + 2n = 2n\).

Key Concepts

Understanding Common Difference

An arithmetic sequence is a series of numbers where each term increases (or decreases) by a fixed amount that we call the common difference. This is simply the amount you add or subtract to get from one term to the next.

For the sequence given in the exercise, the series is 0, 2, 4, 6, 8, ...

• Each term increases by the same amount, which is 2.
• This consistent increase is what defines our common difference, denoted as "d".

The common difference tells us that if you know one term, you can easily find the next term by adding this difference.
To find the common difference, subtract any term from the next term:

• For instance, 2 - 0 = 2.
• Similarly, for the terms 4 and 2, we have 4 - 2 = 2.

Identifying the common difference is a crucial first step in understanding any arithmetic sequence.

Exploring the nth Term Formula

In an arithmetic sequence, once the common difference is known, we can use the nth term formula of the sequence to find any term in the series without having to list them all.
This formula is \[ a_n = a_0 + n \cdot d \]

• Here, \( a_n \) represents the nth term.
• \( a_0 \) is the first term of the sequence, in this case, 0.
• \( n \) is the position number in the sequence.
• And \( d \) is the common difference, which we've identified as 2.

In this specific sequence:

• Putting \( a_0 = 0 \), \( d = 2 \), the formula becomes \( a_n = 0 + n \cdot 2 = 2n \).

This formula allows us to easily compute any term in the sequence. For example, to find the 5th term (when \( n = 5 \)), plug it into the formula to get \( a_5 = 2 \times 5 = 10 \). It's that straightforward!

Using Sequence Pattern Recognition

Recognizing patterns in a sequence is a fundamental skill in mathematics. It allows us to identify if a sequence is arithmetic, geometric, or something else. For arithmetic sequences, being able to spot a consistent change between terms indicates a uniform pattern characteristic of these sequences.
In the example sequence 0, 2, 4, 6, 8, ..., the pattern is clear.

• Each term is simply adding the common difference to the previous term.
• This consistent increment indicates the arithmetic nature of the sequence.

To effectively recognize patterns:

• Start by looking at the difference between consecutive terms.
• If the difference is constant, the sequence is likely arithmetic, as is the case here.

With this pattern recognition, not only can you quickly determine the type of sequence but also apply the appropriate formulas to find terms, helping you solve numerous sequential problems efficiently.