Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence defined recursively. $$a_{0}=1, a_{1}=3, a_{k}=a_{k-2}+a_{k-1}$$

Step-by-Step Solution

Answer

The first five terms of the sequence are: 1, 3, 4, 7, 11.

Step 1: Identify Initial Terms

The sequence provides two initial terms \(a_{0}=1\) and \(a_{1}=3\). These will be the first two terms of the sequence.

Step 2: Apply Recurrence Relation for the 3rd term

Now, use the recursive formula \(a_{k}=a_{k-2}+a_{k-1}\) to find the third term \(a_{2}\). So, \(a_{2} = a_{0} + a_{1} = 1 + 3 = 4.\)

Step 3: Apply Recurrence Relation for the 4th term

The fourth term \(a_{3}\) is the sum of the second and the third terms. So, \(a_{3} = a_{1} + a_{2} = 3 + 4 = 7.\)

Step 4: Apply Recurrence Relation for the 5th term

The fifth term \(a_{4}\) is the sum of the third and the fourth terms. So, \(a_{4} = a_{2} + a_{3} = 4 + 7 = 11.\)

Key Concepts

Initial terms

The initial terms are the foundation of a recursive sequence. In our exercise, the initial terms are given as \(a_{0}=1\) and \(a_{1}=3\). These are the starting points for calculating the subsequent terms using the recurrence relation.

Understanding initial terms is simple if you think of them as the first building blocks or 'seeds' of the sequence. Without these terms, you would not be able to determine any other terms that follow. Think of them as the first chapter of a book, essential to understand the rest of the story.

When working with sequences, always make sure to carefully note the initial terms as given in the problem. They have already been calculated and fixed, so you don't need any formula to find them.

• Serving as the foundation
• Crucial for future calculations
• Given directly in the problem statement

Recurrence relation

The recurrence relation is a crucial formula used to find each subsequent term in a recursive sequence. In our exercise, the recurrence relation is defined as \(a_{k}=a_{k-2}+a_{k-1}\). This means that any term in the sequence can be found by summing the two previous terms.

Consider it as a mathematical recipe. If you know the ingredients (the previous terms), the recurrence relation instructs you on how to mix them to find the next term. This makes recursive sequences both simple and complex, as the beauty lies in following a simple rule to generate terms that may not initially appear predictable.

To master this concept, practice plugging in different previous terms into the relation. Once you do this, the pattern and logic behind the sequence will become clearer. It is like solving a puzzle: knowing how each piece fits into the next.

• Formula-driven approach
• Sums previous terms
• Crucial for sequence progression

Sequence terms

Sequence terms are the result of applying the initial terms and the recurrence relation. In our example, the first five sequence terms are derived as follows:
- \(a_{0} = 1\) (initial term)
- \(a_{1} = 3\) (initial term)
- \(a_{2} = 4\) (calculated using \(a_{0} + a_{1}\))
- \(a_{3} = 7\) (calculated using \(a_{1} + a_{2}\))
- \(a_{4} = 11\) (calculated using \(a_{2} + a_{3}\))

Understanding sequence terms is like following a chain where one link leads to the next. Each term involves a progression from the known values (initial terms) through the use of the recurrence relation.

Once you have applied the recurrence relation, you will notice that sequence terms can grow quickly and develop intricate relationships. These terms serve not only as numbers but as the unfolding story of the sequence. Mastery of this topic requires practice and attention to detail, ensuring each term follows logically from its predecessors.

• Result of computations
• Evaluated using initial terms and relation
• Represents sequence progression