Problem 11 - Thomas Calculus

Question

Each of Exercises \(7-12\) gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. $$ a_{1}=a_{2}=1, \quad a_{n+2}=a_{n+1}+a_{n} $$

Step-by-Step Solution

Answer

The first ten terms are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Step 1: Understand the given sequence

The problem provides a recursive sequence starting with initial terms \(a_1 = 1\) and \(a_2 = 1\), and the recursion formula \(a_{n+2} = a_{n+1} + a_{n}\). We need to apply the recursion formula to generate subsequent terms.

Step 2: Compute the third term

Using the recursion formula, compute \(a_3\):\[ a_3 = a_2 + a_1 = 1 + 1 = 2 \]

Step 3: Compute the fourth term

Using the recursion formula, compute \(a_4\):\[ a_4 = a_3 + a_2 = 2 + 1 = 3 \]

Step 4: Compute the fifth term

Using the recursion formula, compute \(a_5\):\[ a_5 = a_4 + a_3 = 3 + 2 = 5 \]

Step 5: Compute the sixth term

Using the recursion formula, compute \(a_6\):\[ a_6 = a_5 + a_4 = 5 + 3 = 8 \]

Step 6: Compute the seventh term

Using the recursion formula, compute \(a_7\):\[ a_7 = a_6 + a_5 = 8 + 5 = 13 \]

Step 7: Compute the eighth term

Using the recursion formula, compute \(a_8\):\[ a_8 = a_7 + a_6 = 13 + 8 = 21 \]

Step 8: Compute the ninth term

Using the recursion formula, compute \(a_9\):\[ a_9 = a_8 + a_7 = 21 + 13 = 34 \]

Step 9: Compute the tenth term

Using the recursion formula, compute \(a_{10}\):\[ a_{10} = a_9 + a_8 = 34 + 21 = 55 \]

Step 10: Compile the first ten terms

The first ten terms of the sequence, following the recursion formula from the initial values, are: \\(1, 1, 2, 3, 5, 8, 13, 21, 34, 55\).

Key Concepts

Fibonacci sequence

The Fibonacci sequence is a famous series of numbers known for its recursive nature. It starts with two initial terms, both usually set to 1, and each subsequent term is the sum of the two preceding ones.
The sequence begins as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on.
This pattern continues infinitely, where each number in the series is the sum of the two numbers before it.

• Fibonacci numbers are found in nature, such as in the arrangement of leaves on a stem or the branching of trees.
• It has applications in computer algorithms, financial markets, and even art due to its unique mathematical properties.

recursion formula

A recursion formula defines a sequence based on preceding terms. It provides a means to construct a sequence from its initial values: you compute all future terms using previously calculated ones.
For the Fibonacci sequence, the recursion formula is \(a_{n+2} = a_{n+1} + a_{n}\).
This equation instructs us that to find the next term, sum the last two known terms.

• Recursive formulas are commonly used in mathematics and computer science for solving problems where a simple iterative approach is efficient.
• While recursion is straightforward for humans to grasp, it often requires careful implementation to optimize computational resources in programming.

sequence terms calculation

Calculating the terms of a sequence using a recursive formula is an iterative process, ideally suited for sequences like the Fibonacci sequence.
Start with the known initial terms, and apply the recursion formula to find each successive term.
For example:

• Start with \(a_1 = 1\) and \(a_2 = 1\).
• Use the formula \(a_{n+2} = a_{n+1} + a_{n}\) to find \(a_3 = a_2 + a_1 = 2\).
• Continue applying the formula for each next term, leading to the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

By following these steps, you see the beauty of recursive sequences as they build upon themselves to create intricate patterns.

initial conditions in sequences

Initial conditions in sequences are the starting values you need to have in order to apply a recursion formula. These initial values set the foundation of the sequence and allow the recursive formula to generate further terms.
In our Fibonacci exercise, the initial conditions were \(a_1 = 1\) and \(a_2 = 1\).
Without these, the recursion formula \(a_{n+2} = a_{n+1} + a_{n}\) cannot produce any subsequent terms.

• Initial conditions are crucial as they uniquely determine the sequence generated by a recursive formula.
• Different initial conditions can lead to entirely different sequences even, if you use the same recursion formula.

Always make sure to clearly define these initial conditions before proceeding with calculations to ensure accuracy.