Problem 33 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(5,7,9,11,13\)

Step-by-Step Solution

Answer

The expression for \(a_n\) is \(5 + 2n\).

Step 1: Identify the Pattern

First, observe the sequence given: 5, 7, 9, 11, 13. This sequence seems to grow by the same amount in each step. Check the difference between consecutive terms: 7 - 5 = 2, 9 - 7 = 2, 11 - 9 = 2, 13 - 11 = 2. Clearly, the sequence increases by 2 each time, indicating that this is an arithmetic sequence.

Step 2: Determine the First Term and Common Difference

In an arithmetic sequence, a general term can be found using the formula: \(a_n = a_1 + (n-1) imes d\), where \(a_1\) is the first term and \(d\) is the common difference. Here, the first term \(a_0 = 5\), and the common difference \(d = 2\).

Step 3: Write the General Expression

Using the identified values, write the general expression for the sequence. Substitute \(a_0 = 5\) and \(d = 2\) into the formula: \(a_n = 5 + n imes 2\). Simplify to get \(a_n = 5 + 2n\).

Step 4: Verify the Expression

Check the obtained expression with some terms in the sequence: - For \(n = 0\), \(a_0 = 5 + 2 imes 0 = 5\).- For \(n = 1\), \(a_1 = 5 + 2 imes 1 = 7\).- For \(n = 2\), \(a_2 = 5 + 2 imes 2 = 9\).- For \(n = 3\), \(a_3 = 5 + 2 imes 3 = 11\).- For \(n = 4\), \(a_4 = 5 + 2 imes 4 = 13\).The values match, confirming that \(a_n = 5 + 2n\) is correct.

Key Concepts

Sequence Patterns

In mathematics, a sequence is an ordered list of numbers arranged according to a specific rule or pattern. When you explore patterns in sequences, it becomes easier to understand and predict future elements. Sequences can be arithmetic, geometric, or neither. When examining the list of numbers 5, 7, 9, 11, 13, we must identify the type of sequence it represents to better establish a formula.
Noticing the consistent difference between each number, we can see this is an arithmetic sequence.

• Each term has been created by adding a fixed number to the previous term rather than multiplying or using a different pattern.
• For example, adding 2 to 5 gives 7, adding 2 to 7 gives 9, and so forth.

Observing and understanding these sequence patterns help us understand how sequences operate and in designing the formulas that describe them.

Common Difference

In an arithmetic sequence, the 'common difference' is a vital concept. It's the constant value by which consecutive terms in the sequence increase or decrease.
The given sequence 5, 7, 9, 11, 13 shows a repeated addition of 2 between every adjacent term.

• The difference between 7 and 5 is 2.
• The difference between 9 and 7 is also 2.
• This consistent gap is called the common difference, represented by the variable \( d \).

Knowing the common difference helps us craft formulas for predicting future terms, as this steady increase or decrease forms the backbone of the arithmetic sequence.

General Term Formula

The general term formula of an arithmetic sequence provides a straightforward way to find any term in the sequence without listing all preceding terms. It's expressed as:
\[a_n = a_0 + n imes d\] where:

• \( a_n \) is the \( n^{th} \) term.
• \( a_0 \) is the first term.
• \( d \) is the common difference.

For the sequence 5, 7, 9, 11, 13:

• The first term, \( a_0 \), is 5.
• The common difference, \( d \), is 2.

Thus, the formula becomes:
\[a_n = 5 + n imes 2\]This simple yet powerful formula allows you to calculate any term without building the whole sequence. Confirming this formula with substituted values for \( n \) will match results such as 5, 7, 9, validating the formula's accuracy.