Problem 39 - Calculus for Biology and Medicine

Question

Use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=a_{n}-\frac{1}{a_{n}}\) and \(a_{0}=3\), find \(a_{11}\).

Step-by-Step Solution

Answer

In the spreadsheet, cell A12 contains the value of \(a_{11}\).

Step 1: Understand the Recursive Formula

The sequence is defined by the recursive formula \(a_{n+1} = a_{n} - \frac{1}{a_{n}}\) with an initial term of \(a_{0} = 3\). This means each term is calculated based on the preceding term.

Step 2: Set Up the Spreadsheet

In a spreadsheet, enter the initial term \(a_0 = 3\) in cell A1. This is where the computation starts.

Step 3: Calculate the First Few Terms

For cell A2, use the formula \(= A1 - \frac{1}{A1}\) to calculate the next term \(a_1\). Drag this formula down from A2 to fill subsequent cells until A12 to find \(a_{11}\).

Step 4: Verify Each Step

Check each computed term for potential errors by confirming the spreadsheet's calculations with manual calculations of a few terms.

Step 5: Locate the 11th Term

Once the cells are filled, locate the value in cell A12. This value represents \(a_{11}\), as the cell positions account for the initial term starting at A1.

Key Concepts

Spreadsheet Calculations

Using a spreadsheet for calculations can simplify and automate tasks in mathematics, especially when dealing with recursive sequences. A recursive sequence requires you to compute each term based on the previous one, which can be tedious if done manually for many terms. Spreadsheets improve this process by allowing you to write a formula once and apply it down a column to see all the results. This eliminates repetitive calculation and minimizes the risk of error.

To use a spreadsheet effectively for recursive calculations:

• Start by entering the initial term in a cell. For our exercise, this was done in cell A1 with value 3.
• In the adjacent cell, input the recursive formula, referencing the previous cell to calculate the next term.
• Drag the fill handle down the column to automatically apply the formula to subsequent cells.

Spreadsheets also provide easy error-checking. You can verify your results by comparing spreadsheet output with manual calculations of a few terms. This gives you confidence in the accuracy of your data.

Recursive Formula

A recursive formula helps calculate the terms of a sequence by using the preceding term. It is a pattern where each term builds upon the last, thus having a dependency chain stretching from the initial term. In our specific recursive formula, each term is defined by the equation \( a_{n+1} = a_{n} - \frac{1}{a_{n}} \). Here, the next term \( a_{n+1} \) is a function of the prior term \( a_{n} \).

Key aspects of recursive formulas include:

• **Base Case:** This is where the sequence begins. For this exercise, our base case is \( a_0 = 3 \).
• **Recurrence Relation:** This defines how any term in the sequence relates to previous terms. Our recurrence relation is \( a_{n+1} = a_{n} - \frac{1}{a_{n}} \).
• **Dependency on preceding terms:** Each term relies on the computation of previous terms, forming a chain linking back to the initial term.

By understanding and applying the recursive formula, you can simplify complex problems and perform iterative calculations efficiently.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers governed by specific rules. They can be defined explicitly, where each term is defined separately by a formula, or recursively, where each term depends on its predecessors. The sequence type we're focusing on involves recursive calculations, providing a dynamic way of understanding how numbers interact sequentially.

Features of sequences:

• **Order Matters:** Unlike sets, sequences are ordered, meaning the position of each term is significant.
• **Infinite or Finite:** Sequences can go on indefinitely, or have a finite number of terms, as determined by the problem constraints.
• **Rules for Generation:** The terms are generated by specific rules or formulas, such as recursive relationships.

Mathematical sequences are integral to various areas of mathematics and science, serving as foundational elements in understanding patterns, growth, and other phenomena. They are not only useful academically but also in practical applications, such as calculating interest rates and predicting patterns.