Problem 44 - Algebra and Trigonometry

Question

A sequence is defined recursively. List the first five terms. \(a_{1}=-1 ; \quad a_{2}=1 ; \quad a_{n}=a_{n-2}+n a_{n-1}\)

Step-by-Step Solution

Answer

-1, 1, 2, 9, 47

Step 1: Identify the Initial Terms

Given the initial terms of the sequence: \(a_{1} = -1\) \(a_{2} = 1\)

Step 2: Calculate the Third Term

Use the recursive formula \(a_{n} = a_{n-2} + n a_{n-1}\) to find \(a_{3}\): \(a_{3} = a_{1} + 3a_{2} = -1 + 3(1) = -1 + 3 = 2\)

Step 3: Calculate the Fourth Term

Next, use the formula to find \(a_{4}\): \(a_{4} = a_{2} + 4a_{3} = 1 + 4(2) = 1 + 8 = 9\)

Step 4: Calculate the Fifth Term

Lastly, find \(a_{5}\) using the same recursive formula: \(a_{5} = a_{3} + 5a_{4} = 2 + 5(9) = 2 + 45 = 47\)

Step 5: Final Step: List the First Five Terms

Based on the calculations, the first five terms of the sequence are: \(a_{1} = -1,\) \(a_{2} = 1,\) \(a_{3} = 2,\) \(a_{4} = 9,\) \(a_{5} = 47\)

Key Concepts

Understanding Initial Terms

To fully understand recursively defined sequences, we need to start with the initial terms. These are the first values in the sequence that are provided to us. In this exercise, the initial terms are given as:

- The first term, \(a_{1} = -1\)
- The second term, \(a_{2} = 1\)

These terms serve as the building blocks for finding the subsequent terms in the sequence. Without these starting points, it would be impossible to calculate any further terms. Think of the initial terms like the foundation of a building; they support everything that comes after.

Decoding Recursive Formulas

A recursive formula is like a recipe that tells us how to find each term in the sequence using the previous terms. In this problem, the recursive formula provided is:

\[a_{n} = a_{n-2} + n a_{n-1}\]

This formula implies that to find the term \(a_{n}\), we need:
- The term two places before it, \(a_{n-2}\)
- The term right before it, \(a_{n-1}\), multiplied by \(n\) (the position of the term in the sequence)

This kind of formula is useful because it reduces the complexity of finding each term step-by-step, linking each term to its predecessors.

Sequence Calculation

Now, we use the initial terms and the recursive formula to calculate the sequence, as shown in the steps:

1. Find \(a_{3}\):
\(a_{3} = a_{1} + 3a_{2} = -1 + 3(1) = -1 + 3 = 2\)

2. Find \(a_{4}\):
\(a_{4} = a_{2} + 4a_{3} = 1 + 4(2) = 1 + 8 = 9\)

3. Find \(a_{5}\):
\(a_{5} = a_{3} + 5a_{4} = 2 + 5(9) = 2 + 45 = 47\)

So, the first five terms of the sequence are: \(a_{1} = -1, a_{2} = 1, a_{3} = 2, a_{4} = 9, a_{5} = 47\). Each term builds on the previous ones, showing the power of recursion in breaking down complex calculations.

The Role of Algebra

Algebra plays a significant role in understanding and solving recursive sequences. It provides the tools and symbols we need to manipulate the recursive formulas and terms. By understanding algebraic expressions and operations, we can:

- Substitute values into formulas
- Perform arithmetic operations like addition and multiplication
- Simplify complex equations

In this exercise, algebra helped us take the initial terms and recursive formula to calculate and simplify each step. This made it easier to enumerate the terms of the sequence accurately.