Problem 88 - Precalculus: Mathematics for Calculus

Question

Find the first ten 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 ten terms are 1, 1, 2, 3, 3, 4, 5, 5, 6, 6. The sequence is unique due to its recursive index calculation.

Step 1: Initialize the First Two Terms

The sequence starts with the given initial terms: \(a_1 = 1\) and \(a_2 = 1\). These values are necessary for the recursive computation of subsequent terms, as they serve as the basis for the rest of the sequence.

Step 2: Compute the Third Term

To find \(a_3\), use the formula \(a_n = a_{n-a_{n-1}} + a_{n-a_{n-2}}\):Given that \(a_2=1\), substitute into the formula:\[a_3 = a_{3-a_2} + a_{3-a_1} = a_2 + a_2 = 1 + 1 = 2.\]

Step 3: Compute the Fourth Term

Now calculate \(a_4\) using the values obtained:\[a_4 = a_{4-a_3} + a_{4-a_2} = a_2 + a_3 = 1 + 2 = 3.\]

Step 4: Compute the Fifth Term

Continue with the pattern for \(a_5\):\[a_5 = a_{5-a_4} + a_{5-a_3} = a_2 + a_3 = 1 + 2 = 3.\]

Step 5: Compute the Sixth Term

Find \(a_6\) as follows:\[a_6 = a_{6-a_5} + a_{6-a_4} = a_3 + a_3 = 2 + 2 = 4.\]

Step 6: Compute the Seventh Term

Determine \(a_7\):\[a_7 = a_{7-a_6} + a_{7-a_5} = a_3 + a_4 = 2 + 3 = 5.\]

Step 7: Compute the Eighth Term

Calculate \(a_8\):\[a_8 = a_{8-a_7} + a_{8-a_6} = a_3 + a_4 = 2 + 3 = 5.\]

Step 8: Compute the Ninth Term

Next, find \(a_9\):\[a_9 = a_{9-a_8} + a_{9-a_7} = a_4 + a_5 = 3 + 3 = 6.\]

Step 9: Compute the Tenth Term

Finally, calculate \(a_{10}\):\[a_{10} = a_{10-a_9} + a_{10-a_8} = a_4 + a_5 = 3 + 3 = 6.\]

Step 10: Summarize the Sequence

The first ten terms of the sequence are:\[1, 1, 2, 3, 3, 4, 5, 5, 6, 6.\]This recursive sequence differs because the computation of each term depends on previously calculated terms to determine the indices of terms to sum, which is atypical compared to sequences where terms are dependent on fixed-index positions.

Key Concepts

Understanding Sequence

A sequence is a list of numbers arranged in a specific order. Each number in the list is called a term. The sequence typically follows a rule that defines how each term is determined based on its position in the sequence. For instance, in arithmetic or geometric sequences, a fixed pattern allows easy calculation of any given term. The complexity of a sequence can vary, with some having straightforward numerical progressions, while others require more complex calculations.

• Terms: These are the individual elements of the sequence. For example, in the sequence 1, 2, 3, the numbers 1, 2, and 3 are terms.
• Index: This refers to the position of a term in the sequence, often denoted by 'n'. In our sequence, the position is used to determine which terms calculate the next one.
• Rule: The specific calculation or pattern needed to generate the sequence. It indicates how to go from one term to the next.

In complex sequences, like the one from the exercise, the relationship among terms can be unique, making it crucial to correctly apply the defining rule that connects the terms.

Exploring Recurrence Relation

A recurrence relation is an equation that expresses each term of the sequence as a function of the preceding terms. It defines the sequence not with a single overarching formula, but through relationships linking consecutive terms. In simpler cases, such as the Fibonacci sequence, each term is the sum of the two preceding terms.

In our exercise, the recurrence relation for a sequence is given by:

• For any term starting from the third, each term is computed as: \[ a_n = a_{n-a_{n-1}} + a_{n-a_{n-2}} \] This means that to calculate a term, we don’t directly use the indexing as constant positions like 1 or 2, but rather use information from previous terms to find effective indices.

This complexity highlights how a recurrence relation can create unique sequence behavior different from conventional models. Essentially, rather than a fixed recursive blueprint, our sequence dynamically adjusts indices based on earlier terms.

Using Mathematical Induction

Mathematical induction is a technique used to prove the validity of statements, formulas, or sequences. It’s a bit like falling dominoes – pushing one sets off a chain reaction that proves the entire setup.

To apply mathematical induction, it involves two key steps:

• Base Case: Establish the statement’s truth for the initial value, usually the first term. Here, you show that the beginning of the sequence holds true as per the rule, such as verifying given initial terms like \(a_1=1\) and \(a_2=1\).
• Inductive Step: Assume the statement holds for 'k'. Then, prove it for 'k+1'. This step ensures if it works for one arbitrary position, it works for the next. This leap from 'k' to 'k+1' is the crucial step in tying together the sequence.

In our exercise, each computation carries forward by using previously proven steps, validating new terms. This process confirms the sequence logic holds for successive terms, forming a continuous proof trail that secures the sequence’s integrity throughout its counting order.