Problem 97 - Calculus for Biology and Medicine

Question

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=\frac{a_{n}}{1+a_{n}}, a_{0}=1 $$

Step-by-Step Solution

Answer

The terms are \( a_1 = \frac{1}{2} \), \( a_2 = \frac{1}{3} \), \( a_3 = \frac{1}{4} \), \( a_4 = \frac{1}{5} \), and \( a_5 = \frac{1}{6} \).

Step 1: Calculate a_1

Start with the initial value, \( a_0 = 1 \). Using the recursive formula, compute \( a_1 \):\[ a_1 = \frac{a_0}{1 + a_0} = \frac{1}{1 + 1} = \frac{1}{2}. \]

Step 2: Calculate a_2

Using the value of \( a_1 \) from the previous step, compute \( a_2 \):\[ a_2 = \frac{a_1}{1 + a_1} = \frac{\frac{1}{2}}{1 + \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{3}{2}} = \frac{1}{3}. \]

Step 3: Calculate a_3

Using the value of \( a_2 \), compute \( a_3 \):\[ a_3 = \frac{a_2}{1 + a_2} = \frac{\frac{1}{3}}{1 + \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{4}{3}} = \frac{1}{4}. \]

Step 4: Calculate a_4

Using the value of \( a_3 \), compute \( a_4 \):\[ a_4 = \frac{a_3}{1 + a_3} = \frac{\frac{1}{4}}{1 + \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{5}{4}} = \frac{1}{5}. \]

Step 5: Calculate a_5

Using the value of \( a_4 \), compute \( a_5 \):\[ a_5 = \frac{a_4}{1 + a_4} = \frac{\frac{1}{5}}{1 + \frac{1}{5}} = \frac{\frac{1}{5}}{\frac{6}{5}} = \frac{1}{6}. \]

Key Concepts

Understanding the Recursive Formula

A recursive formula is a way to determine the elements of a sequence using previous elements. It's like following a set of instructions where the result of one step leads to the next. In mathematics, recursive formulas are often used to model situations where each new term depends on one or more preceding terms.
For our given problem, the recursive formula is:

• This formula describes how to find \(a_{n+1}\), the next term in the sequence, by using \(a_n\), the current term.
• The operation \(\frac{a_n}{1 + a_n}\) means we divide the current term \(a_n\) by one more than itself.

This recursive setup tells us exactly how to step from one term to the next, making it a handy tool for solving problems involving sequences.

Establishing the Initial Value

The initial value of a sequence is the starting point from which all other elements are calculated using the recursive formula. It's essential because without a starting point, the recursive formula cannot generate any subsequent terms. In our exercise, the initial value given is:

• \(a_0 = 1\)

This means that before we start applying the recursive formula, the very first term of the sequence is 1. This initial value feeds into the recursive formula, kickstarting the entire sequence computation process.

Carrying Out Sequence Computation

Sequence computation is the process of determining all the terms in a sequence by using the recursive formula and initial value. Each term is derived from the previous one following the strict rule established by the recursive formula. Let's see how the sequence is computed from \(n=1\) to \(n=5\) utilizing our recursive formula:

• For \(a_1\): Start with \(a_0 = 1\), and apply the formula: \(a_1 = \frac{1}{1 + 1} = \frac{1}{2}\)
• For \(a_2\): Use \(a_1 = \frac{1}{2}\), so: \(a_2 = \frac{\frac{1}{2}}{1 + \frac{1}{2}} = \frac{1}{3}\)
• For \(a_3\): With \(a_2 = \frac{1}{3}\), compute: \(a_3 = \frac{\frac{1}{3}}{1 + \frac{1}{3}} = \frac{1}{4}\)
• For \(a_4\): Given \(a_3 = \frac{1}{4}\), find: \(a_4 = \frac{\frac{1}{4}}{1 + \frac{1}{4}} = \frac{1}{5}\)
• For \(a_5\): Using \(a_4 = \frac{1}{5}\), calculate: \(a_5 = \frac{\frac{1}{5}}{1 + \frac{1}{5}} = \frac{1}{6}\)

Through this methodical process, we seamlessly compute the terms of the sequence one after another, showcasing the power of recursion in solving complex problems.