Problem 82 - Algebra and Trigonometry

Question

A Different Type of Recursion Find the first 10 terms of the sequence defined by $$a_{n}=a_{n-a_{n-1}}+a_{n-a_{n-2}}$$ with $$a_{1}=1 \quad \text { and } \quad a_{2}=1$$ How is this recursive sequence different from the others in this section?

Step-by-Step Solution

Answer

The first 10 terms are 1, 1, 2, 3, 3, 4, 5, 5, 6, 6. This sequence uses previous terms to find indices that dictate summation.

Step 1: Understand the Sequence Definition

The sequence is defined recursively by \( a_n = a_{n-a_{n-1}} + a_{n-a_{n-2}} \). We start with initial values \( a_1 = 1 \) and \( a_2 = 1 \). Unlike simple arithmetic or geometric sequences, this sequence's nth term depends on the indices determined by previous terms themselves.

Step 2: Calculate Initial Terms

We start with the initial terms \( a_1 = 1 \) and \( a_2 = 1 \).

Step 3: Calculate the Third Term

To find \( a_3 \), apply the formula: \( a_3 = a_{3-a_2} + a_{3-a_1} = a_{3-1} + a_{3-1} = a_2 + a_2 \). Therefore, \( a_3 = 1 + 1 = 2 \).

Step 4: Calculate the Fourth Term

For \( a_4 \), use \( a_4 = a_{4-a_3} + a_{4-a_2} = a_{4-2} + a_{4-1} = a_2 + a_3 \), yielding \( a_4 = 1 + 2 = 3 \).

Step 5: Calculate the Fifth Term

Determine \( a_5 \): \( a_5 = a_{5-a_4} + a_{5-a_3} = a_{5-3} + a_{5-2} = a_2 + a_3 \). Thus, \( a_5 = 1 + 2 = 3 \).

Step 6: Calculate the Sixth Term

Find \( a_6 \) using the formula: \( a_6 = a_{6-a_5} + a_{6-a_4} = a_{6-3} + a_{6-3} = a_3 + a_3 \), which gives \( a_6 = 2 + 2 = 4 \).

Step 7: Calculate the Seventh Term

To find \( a_7 \), compute: \( a_7 = a_{7-a_6} + a_{7-a_5} = a_{7-4} + a_{7-3} = a_3 + a_4 \). Thus, \( a_7 = 2 + 3 = 5 \).

Step 8: Calculate the Eighth Term

Find \( a_8 \) by using: \( a_8 = a_{8-a_7} + a_{8-a_6} = a_{8-5} + a_{8-4} = a_3 + a_4 \), resulting in \( a_8 = 2 + 3 = 5 \).

Step 9: Calculate the Ninth Term

For \( a_9 \), calculate: \( a_9 = a_{9-a_8} + a_{9-a_7} = a_{9-5} + a_{9-5} = a_4 + a_4 \). Therefore, \( a_9 = 3 + 3 = 6 \).

Step 10: Calculate the Tenth Term

Find \( a_{10} \) with: \( a_{10} = a_{10-a_9} + a_{10-a_8} = a_{10-6} + a_{10-5} = a_4 + a_5 \), which gives \( a_{10} = 3 + 3 = 6 \).

Step 11: Review and Summarize

The first ten terms of the sequence are 1, 1, 2, 3, 3, 4, 5, 5, 6, and 6. The recursive nature of the sequence and dependence of each term on indices modified by previous terms differentiates this sequence from more traditional recursive sequences.

Key Concepts

Sequence Definition

A sequence is a set of numbers arranged in a specific order. It's like a list where each number in the list has a specific place. In mathematics, sequences are often defined by a rule or formula that tells us how to find each number, or term, in the sequence.

• For arithmetic sequences, each term is found by adding a constant to the previous term.
• For geometric sequences, each term is found by multiplying the previous term by a constant.
• Recursive sequences, like the one in our exercise, use previous terms and specific rules to define new terms.

Since sequences can be defined in various ways, understanding the underlying rule is crucial to accurately generating the terms.

Initial Terms

Initial terms are the starting point of any sequence. They are the first few numbers in the sequence that are known or given before applying any formulas or rules. The initial terms act as a foundation for building the rest of the sequence, especially in recursive sequences.In our example, the sequence begins with initial terms:

• \( a_1 = 1 \)
• \( a_2 = 1 \)

The values of these initial terms are needed to start utilizing the recursive formula that defines later terms. Without these starting terms, calculating the rest of the sequence wouldn't be possible.

Recursive Formula

A recursive formula provides a method for finding each term in the sequence by using the preceding terms. It is like a mathematical procedure repeated over and over. This type of formula requires knowing the initial terms to get started.The given recursive formula for our unique sequence is:\[a_n = a_{n-a_{n-1}} + a_{n-a_{n-2}}\]This formula implies that, to find any term \(a_n\), we must reference terms that are positioned based on previous terms' values. It's a bit tricky because, unlike simpler sequences, it incorporates changes to the position from which terms are taken.

Sequence Terms

Sequence terms are the individual elements or numbers in the sequence. They are generated by following the rule or formula specific to that sequence. For the sequence in this exercise, the first ten sequence terms have been calculated as follows: 1. \( a_1 = 1 \)2. \( a_2 = 1 \)3. \( a_3 = 2 \)4. \( a_4 = 3 \)5. \( a_5 = 3 \)6. \( a_6 = 4 \)7. \( a_7 = 5 \)8. \( a_8 = 5 \)9. \( a_9 = 6 \)10. \( a_{10} = 6 \)These values are derived using the recursive formula, where each term result influences the subsequent calculations. This interconnected dependency makes such sequences interesting and complex to solve.