Problem 30 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(\frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11}, \ldots\)

Step-by-Step Solution

Answer

\(a_n = \frac{n+1}{2n+3}\)

Step 1: Identify the Pattern in the Numerator

The sequence of numerators is 1, 2, 3, 4, 5, ... This is an arithmetic sequence where the pattern increases by 1 each time. Thus, the general formula for the numerator is simply the index + 1, or more specifically, the numerator for term \(a_n\) is \(n+1\).

Step 2: Identify the Pattern in the Denominator

The sequence of denominators is 3, 5, 7, 9, 11, ... This is an arithmetic sequence where each term increases by 2. You could express the nth term of the denominator as starting from 3 and adding \(2n\) to it. Therefore, the denominator for term \(a_n\) is \(2n + 3\).

Step 3: Write the General Expression for \(a_n\)

Combine the observations from Steps 1 and 2 to formulate the sequence term expression. Using the numerator from Step 1 and the denominator from Step 2, the expression for the general term \(a_n\) is \(\frac{n+1}{2n+3}\).

Key Concepts

Numerator Pattern

When analyzing a sequence, it's important to detect any underlying patterns, particularly in the numerator when dealing with arithmetic sequences of fractions. In the given exercise, the sequence of numerators is 1, 2, 3, 4, 5, which follows a consistent increase. Each number in this list increases by 1 from the previous. This regular increase indicates an arithmetic sequence.

The pattern for the numerator can be expressed with a simple formula. If you consider each index (starting from zero), each term in the numerator is simply the index plus one. So, for the nth term of the series, the numerator is given by the expression:

• Numerator for term 1 is 1 because it's 0 + 1.
• Numerator for term n is simply n + 1.

Recognizing this pattern allows us to predict any future term's numerator by simply knowing its position in the sequence.

Denominator Pattern

To fully understand a sequence with fractions, we also need to identify the pattern within the denominators. In this case, the denominator sequence starts as 3, 5, 7, 9, 11. Similar to the numerators, the denominators also form an arithmetic sequence, but with a different step. This sequence increases by 2 for each term, which makes it straightforward to identify a formula.

The nth term of the denominator starts at the number 3 and each subsequent number increases by 2. Therefore, the general formula for calculating any term's denominator is:

• For the first term, the denominator is 3, which is represented as 1 * 2 + 1.
• The formula for subsequent terms is 2n + 3.

Knowing this pattern, it's easy to compute the denominator for any term in the sequence without having to list all previous terms.

General Term Formula

Once both the numerator and the denominator patterns are established, the final step is to put them together to determine the general term formula for the sequence. In this particular problem, by combining our insights about the numerators and denominators, we find a concise representation of the sequence terms.

For the nth term in the sequence:

• The numerator is given by n + 1.
• The denominator is determined by 2n + 3.

Thus, the general term, denoted as \(a_n\), can be written as:\[a_{n} = \frac{n+1}{2n+3}\]This formula provides a quick way to calculate any term in the sequence without writing out all previous fractions. Understanding how to derive such a formula is crucial for efficiently dealing with arithmetic sequences of fractions.