Problem 16 - Algebra and Trigonometry

Question

Find the first five terms of the given recursively defined sequence. \(a_{n}=a_{n-1}+a_{n-2}+a_{n-3} \quad\) and \(\quad a_{1}=a_{2}=a_{3}=1\)

Step-by-Step Solution

Answer

The first five terms are 1, 1, 1, 3, and 5.

Step 1: Understand the Recursive Formula

The sequence is recursively defined as \( a_n = a_{n-1} + a_{n-2} + a_{n-3} \). This means that each term is the sum of the previous three terms. The initial terms given are \( a_1 = 1 \), \( a_2 = 1 \), and \( a_3 = 1 \).

Step 2: Calculate the Fourth Term

To find \( a_4 \), use the recursive formula: \( a_4 = a_3 + a_2 + a_1 \). Substitute the known values: \( a_4 = 1 + 1 + 1 = 3 \).

Step 3: Calculate the Fifth Term

To find \( a_5 \), apply the recursive formula: \( a_5 = a_4 + a_3 + a_2 \). Substitute the known values: \( a_5 = 3 + 1 + 1 = 5 \).

Step 4: Compile the First Five Terms

Now we have found the first five terms of the sequence: \( a_1 = 1 \), \( a_2 = 1 \), \( a_3 = 1 \), \( a_4 = 3 \), and \( a_5 = 5 \).

Key Concepts

Algebra

Algebra forms the foundation for understanding recursive sequences like the one in the original exercise. To effectively work through and solve problems involving sequences, one must grasp algebraic expressions and how they define relationships between numbers. In the case of recursive sequences, algebra helps express how each term in a series relates to previous terms. To clarify, consider the expression \( a_n = a_{n-1} + a_{n-2} + a_{n-3} \). This algebraic formula specifies that each term, marked by \( a_n \), is calculated by summing the three terms that precede it: \( a_{n-1} \), \( a_{n-2} \), and \( a_{n-3} \). Understanding this algebraic relationship is key to deciphering recursive sequences.

Sequence Calculation

Sequence calculation is crucial when dealing with problems involving recursions. By understanding how each term is calculated, you can effectively predict future terms based on previous ones.To calculate terms in this recursive sequence:

• Start with the base terms: \( a_1, a_2, \) and \( a_3 \), all equal to 1.
• Use the formula provided: \( a_n = a_{n-1} + a_{n-2} + a_{n-3} \).

For example, to find \( a_4 \), use the initial three terms: \( a_4 = 1 + 1 + 1 = 3 \). For \( a_5 \), use the terms \( a_4, a_3, \) and \( a_2 \): \( a_5 = 3 + 1 + 1 = 5 \). These calculations form a systematic approach to sequence calculation.

Sequence Terms

Terms in a sequence denote specific values and positions within a series. In a recursively defined sequence, these terms are interconnected, each influenced by its predecessors.The terms are labeled with subscript notation such as \( a_1, a_2, \) and so forth. In the example problem, the first three terms are provided as initial conditions: \( a_1 = a_2 = a_3 = 1 \).The importance of these initial terms is to "launch" the sequence. Without these, further terms cannot be determined. As each new term is calculated, it adds to the sequence, providing a richer pattern of relationships between the numbers.

Mathematical Problem Solving

Problem solving in mathematics involves an approach that combines understanding the problem, applying mathematical concepts, and verifying your results. To tackle the recursive sequence:

• Start by understanding the recursive nature of the formula which dictates how terms are derived.
• Use initial terms effectively to compute subsequent terms using the given rule.
• Verify calculations to ensure each term follows logically from the previous ones.

This structured process not only helps in solving specific problems but also builds analytical skills useful for broader mathematical challenges.