Problem 21 - Calculus for Biology and Medicine

Question

Find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots, a_{5}\) $$ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6} $$

Step-by-Step Solution

Answer

The next four values are \(\frac{6}{7}, \frac{7}{8}, \frac{8}{9}, \frac{9}{10}\).

Step 1: Identify the Sequence Pattern

Observe the values of the sequence given: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}\). Notice that each fraction is in the form \(\frac{n}{n+1}\).

Step 2: Apply the Pattern to Previous Terms

For each term in the sequence, the numerator is the position number starting from 1 and the denominator is the position number plus 1. For example, \(a_1 = \frac{1}{2}\), \(a_2 = \frac{2}{3}\), \(a_3 = \frac{3}{4}\), and so on. This confirms the pattern \(a_n = \frac{n}{n+1}\).

Step 3: Calculate the Next Term Using the Pattern

To find the next term \(a_6\), use the formula \(a_n = \frac{n}{n+1}\). Therefore, \(a_6 = \frac{6}{7}\).

Step 4: Continue the Sequence Calculation

Continue using the formula for the next terms. \(a_7 = \frac{7}{8}\), \(a_8 = \frac{8}{9}\), and \(a_9 = \frac{9}{10}\).

Step 5: Compile the Next Four Terms

The next four values of the sequence are \(\frac{6}{7}, \frac{7}{8}, \frac{8}{9}, \frac{9}{10}\).

Key Concepts

Fraction Sequences

Fraction sequences are a fascinating type of mathematical sequence where each term is expressed as a fraction. They often follow a distinct pattern that makes them predictable and easy to extend. In the given exercise, the sequence

• consists of fractions: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}\).
• When extended, each fraction's value gets closer to 1 as the numerator and denominator become more similar.

The beauty of fraction sequences lies in identifying that pattern. Simply by observing the first few terms, you can deduce how to obtain the subsequent terms. This makes them a powerful example in understanding mathematical concepts like limits and convergence in more advanced mathematics.

Numerator-Denominator Relationships

Understanding numerator-denominator relationships is crucial in analyzing fraction sequences effectively. In this sequence, there is a clear relationship between numerators and denominators:

• Each numerator is greater by 1 than the numerator in the preceding fraction.
• Each denominator is exactly one more than its corresponding numerator.

Notice how this results in each term being of the form \(\frac{n}{n+1}\). This is why our sequence begins with \(\frac{1}{2}\) and continues with \(\frac{2}{3}\), \(\frac{3}{4}\), and so forth. Understanding this relationship will help identify and predict future terms without having to recalculate from scratch each time. By applying this logic, you can quickly derive the future terms in the sequence, knowing that as the sequence progresses, each numerator takes on the value of the previous denominator paired with one, maintaining the fraction's unique identity in the sequence.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers arranged according to a pattern or a rule. In our exercise, the pattern is given by the formula \(a_n = \frac{n}{n+1}\). Here's what makes sequences intriguing:

• A sequence can be defined by identifying the first few terms and then finding a rule that describes all subsequent terms.
• Mathematicians use sequences to explore concepts such as convergence or to define more complex mathematical objects like series or functions.

The primary goal when working with sequences is to discern the rule governing them. This allows for the easy computation of any desired term. In many practical scenarios, sequences are used in calculating interest rates, predicting population growth, or even in constructing algorithms in computer science. Whether simple or complex, understanding sequences is foundational to many areas of mathematical study and application.