Problem 19 - Precalculus: Mathematics for Calculus

Question

A sequence is defined recursively by the given formulas. Find the first five terms of the sequence. \(a_{n}=a_{n-1}+a_{n-2} \quad\) and \(\quad a_{1}=1, a_{2}=2\)

Step-by-Step Solution

Answer

The first five terms are 1, 2, 3, 5, and 8.

Step 1: Understand the Problem

We need to find the first five terms of a sequence defined recursively. We're given the formula \(a_{n} = a_{n-1} + a_{n-2}\) and initial conditions \(a_{1} = 1\) and \(a_{2} = 2\).

Step 2: Calculate the Third Term

Use the recursive formula to find \(a_3\). Substitute the values of \(a_1\) and \(a_2\): \[ a_3 = a_2 + a_1 = 2 + 1 = 3 \]

Step 3: Calculate the Fourth Term

Use the recursive formula to find \(a_4\). Substitute the values of \(a_2\) and \(a_3\): \[ a_4 = a_3 + a_2 = 3 + 2 = 5 \]

Step 4: Calculate the Fifth Term

Use the recursive formula to find \(a_5\). Substitute the values of \(a_3\) and \(a_4\): \[ a_5 = a_4 + a_3 = 5 + 3 = 8 \]

Step 5: Compile the First Five Terms

List the terms that have been calculated: \(a_1 = 1\), \(a_2 = 2\), \(a_3 = 3\), \(a_4 = 5\), and \(a_5 = 8\).

Key Concepts

Fibonacci sequence

The Fibonacci sequence is one of the most famous and intriguing sequences in mathematics. It starts with two given numbers, usually 0 and 1, and each subsequent term is the sum of the two preceding ones. Although in our original exercise, the starting terms are 1 and 2, the basic idea remains the same. Starting with two known values allows us to calculate the terms that follow. This sequence not only stands out for its elegant relationship between numbers but also for its widespread application in nature, art, and algorithm design.

• In natural sciences, the Fibonacci numbers appear in the patterns of leaves, flowers, and even in the branching of trees.
• Artists and architects often find the Fibonacci sequence when designing aesthetically pleasing structures.
• In computer science, algorithms based on this sequence can optimize many calculations.

The sequence grows quite rapidly because each term builds on the previous two, making it all the more fascinating to study and apply.

recursion

Recursion is a fundamental concept in computer science and mathematics, characterized by a process in which a function calls itself to solve a problem. In the context of numerical sequences, recursion provides a powerful technique to define sequences without explicitly listing all the terms. Instead, it uses a rule that refers back to earlier terms.

• A recursive sequence begins with one or more initial terms.
• Each subsequent term is constructed using these initial values according to a specific rule or formula.
• Recursion is particularly useful when dealing with problems that can be broken down into simpler, repeatable tasks.

The recursive definition of a sequence not only simplifies the expression of complex patterns but also aids in understanding the relationship between terms intuitively. Recursive formulas, like the one in our exercise, allow us to determine any term efficiently once the base cases are known.

terms of a sequence

In any mathematical sequence, understanding the roles and relationships of individual terms is crucial. A 'term' in a sequence is simply one of the sequence's elements, and knowing how these elements are constructed is key to grasping the sequence's overall structure. In the context of our task, discovering the first few terms helped us start from initial conditions and journey through the recursive formula.

• The first term, often denoted as \( a_1 \), serves as a starting point.
• Each subsequent term is dependent on one or more of the previous terms via a recursive relationship.
• The placement and value of terms can change relative concepts in mathematics and real-world applications.

Finding the terms of a sequence allows us to see numerical patterns emerge, showing how relationships within data can become visible. This systematic approach eases finding later terms while ensuring those initial conditions are critical stepping stones for any sequence, ensuring we accurately map its entire path.