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

The general term for the sequence is \(a_n = \frac{n+1}{2n+3}\).

Step 1: Observe the pattern

Examine the terms in the sequence: \( \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11}, \ldots \). Notice that the numerators increase by 1: 1, 2, 3, 4, 5, etc. The numerators follow the sequence \(1, 2, 3, 4, 5, \ldots\) which suggests a general term for the numerator as \(n + 1\), where \(n\) starts from 0.

Step 2: Analyze the denominators

Look at the denominators of the sequence: 3, 5, 7, 9, 11, etc. Each denominator increases by 2 from the previous term. The sequence is an arithmetic progression starting from 3 with a common difference of 2. This can be expressed as \(2n + 3\).

Step 3: Form the general term

Now that we have identified expressions for both the numerator and the denominator, combine them to form the general term for \(a_n\). The numerator is \(n + 1\) and the denominator is \(2n + 3\). Therefore, the general expression for \(a_n\) is \(a_n = \frac{n+1}{2n+3}\).

Key Concepts

Arithmetic Progression

The concept of an arithmetic progression is fundamental when analyzing sequences, especially those involving sequences that change at a constant rate. This particular sequence has a term's denominator forming an arithmetic progression. Such a sequence is characterized by a starting number and a consistent interval of increase or decrease between the consecutive terms. Here, the denominators 3, 5, 7, 9, 11, etc., form a sequence where each number is two more than the previous one.

An arithmetic progression (AP) is defined by two elements: the first term and the common difference. In this example, the starting number for the denominators is 3, with a common difference (increment) of 2. This means for any term in the sequence:

• The first term, often called "a", is 3.
• The common difference, often represented as "d", is 2.

The general formula for the nth term of an arithmetic progression is given by:

• \( a_n = a + (n-1) imes d \)

For our sequence, adjusting the formula to start counting from zero (i.e., the 0-th term rather than the 1st), it transforms to:

• \( a_n = 2n + 3 \)

Understanding this pattern helps unravel complex sequences into predictable components.

Numerator and Denominator Patterns

In any sequence, recognizing patterns within the numerators and denominators is crucial for deriving a sequence's formula. Here we have a sequence where both numerator and denominator change with a systematic pattern.

• The numerator pattern is straightforward: it increases by 1 with each step, forming the simple sequence 1, 2, 3, 4, 5, etc. This can be generally described by the formula \( n + 1 \), where \( n \) refers to the position of the term, starting from 0.
• The denominator, as previously identified, follows an arithmetic progression pattern: 3, 5, 7, 9, 11,... with a constant increment of 2. This arithmetic series can be generalized by the formula \( 2n + 3 \) as described in the previous section.

By clearly identifying and understanding these patterns in the numerator and the denominator, it becomes much easier to synthesize the full general term expression that represents the entire sequence accurately.

General Term Formula

A sequence's general term formula brings together all observed patterns into a single expression that defines any term in the sequence. Given the patterns identified in both the numerator and the denominator of our sequence, formulating a general term becomes a straightforward task.

The sequence provided is \( \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11}, \dots \). From this, it's evident that:

• Each numerator aligns with the simple increase sequence: \( n + 1 \).
• Each denominator is described by the arithmetic progression: \( 2n + 3 \).

Hence, the general term formula that captures both the numerator and denominator patterns beautifully is:

• \( a_n = \frac{n + 1}{2n + 3} \)

This general formula allows you to compute the value of any term in the sequence by substituting the term's position \( n \) into the formula. This consolidated approach makes understanding and working with sequences much more manageable and less daunting.