Problem 12 - 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. $$ a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{n} $$

Step-by-Step Solution

Answer

The first ten terms of the sequence are 2, -1, -0.5, 0.5, -1, -2, 2, -1, -0.5, 0.5.

Step 1: Calculate Third Term

We are given that \( a_{1} = 2 \) and \( a_{2} = -1 \). To find \( a_{3} \), use the recursion formula: \( a_{n+2} = a_{n+1} / a_{n} \). Thus, \( a_{3} = a_{2} / a_{1} = -1 / 2 = -0.5 \).

Step 2: Calculate Fourth Term

Using the recursion formula again, \( a_{4} = a_{3} / a_{2} = -0.5 / -1 = 0.5 \).

Step 3: Calculate Fifth Term

Find \( a_{5} \) by calculating \( a_{5} = a_{4} / a_{3} = 0.5 / -0.5 = -1 \).

Step 4: Calculate Sixth Term

Find \( a_{6} \) by calculating \( a_{6} = a_{5} / a_{4} = -1 / 0.5 = -2 \).

Step 5: Calculate Seventh Term

Find \( a_{7} \) by calculating \( a_{7} = a_{6} / a_{5} = -2 / -1 = 2 \).

Step 6: Calculate Eighth Term

Find \( a_{8} \) by calculating \( a_{8} = a_{7} / a_{6} = 2 / -2 = -1 \).

Step 7: Calculate Ninth Term

Find \( a_{9} \) by calculating \( a_{9} = a_{8} / a_{7} = -1 / 2 = -0.5 \).

Step 8: Calculate Tenth Term

Find \( a_{10} \) by calculating \( a_{10} = a_{9} / a_{8} = -0.5 / -1 = 0.5 \).

Key Concepts

Sequence Terms

A sequence is essentially an ordered list of numbers. Each number in this list is called a term. In our problem, we are constructing a sequence based on certain rules.
Our given sequence starts with two terms: \( a_1 = 2 \) and \( a_2 = -1 \). These starting numbers are crucial as they set the base for the entire sequence generated through our recursive process.
Understanding the initial terms is important because each subsequent term relies on these values, combined with the rule given by the recursion formula.

Recursion Formula

Recursion is a method where the solution to a problem depends on solutions to smaller instances of the same problem. In simple terms, a recursion formula in mathematics is a way to build a sequence from the previous terms.
For our sequence, the recursion formula is \( a_{n+2} = \frac{a_{n+1}}{a_n} \). This tells us that to find any term in the sequence from the third term onward, we must divide the previous term by the term before it.
Recursion formulas are useful because they allow us to define a potentially infinite sequence starting from just one or a few terms.

Mathematical Sequences

Mathematical sequences are collections of numbers arranged in a specific order based on a given rule. They can range from simple, predictable sequences to more complex ones like the one in our problem.
Sequences can be infinite or finite, arithmetic, geometric, or of another variety. In our case, we are exploring a sequence defined by division, known as a recursive sequence.
The beauty of sequences lies in their variety and the methods available to analyze and solve them. The recursive sequence here showcases how mathematical sequences can evolve in unexpected yet systematic ways.

Term Calculation

Calculating terms in a recursive sequence involves applying the recursion formula repeatedly. Each term calculated forms the foundation for the next, creating a chain of dependencies.
For our sequence:

• Step 1: \( a_1 = 2 \), \( a_2 = -1 \)
• Step 2: \( a_3 = \frac{-1}{2} = -0.5 \)
• Step 3: \( a_4 = \frac{-0.5}{-1} = 0.5 \)
• And so on...

By repeating the process, using previous terms as inputs, new terms are projected forward in the sequence.
This method is efficient as it transforms a seemingly complex problem into manageable steps, allowing for continued exploration and discovery.