Problem 18 - Algebra and Trigonometry

Question

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=a_{n-1}+a_{n-2}+a_{n-3} \quad \text { and } \quad a_{1}=a_{2}=a_{3}=1 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Sequence

The sequence is defined recursively, meaning each term depends on previous terms. The recursive formula given is \(a_{n} = a_{n-1} + a_{n-2} + a_{n-3}\). Additionally, the initial terms are specified as \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\).

Step 2: Calculating Fourth Term

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

Step 3: Calculating Fifth Term

Use the recursive formula to find \(a_5\): \(a_5 = a_4 + a_3 + a_2 = 3 + 1 + 1 = 5\). Therefore, \(a_5 = 5\).

Step 4: Final Sequence

The first five terms of the sequence are determined as follows: \(a_1 = 1\), \(a_2 = 1\), \(a_3 = 1\), \(a_4 = 3\), \(a_5 = 5\).

Key Concepts

Recursive Formula

A recursive formula is a way to define sequences where each term is expressed using previous terms. This type of formula is useful when the sequence has clear step-by-step dependencies on preceding values. In our exercise, we have the recursive formula:\[ a_n = a_{n-1} + a_{n-2} + a_{n-3} \]This tells us that to find any term in the sequence, say \( a_n \), we need to know the three previous terms: \( a_{n-1} \), \( a_{n-2} \), and \( a_{n-3} \). Such formulas are powerful because they provide a way to extend the sequence indefinitely or as far as needed just by knowing a few starting terms and following the established pattern. Recursive sequences are particularly common in computer science and mathematics because they help in breaking down complex problems into simpler ones.

Initial Terms in Sequences

Initial terms in a sequence are the first few terms that serve as the base to generate subsequent terms using the recursive formula. They are essential for setting the sequence into motion, providing a starting point for calculations. In our given sequence, the initial terms are defined as:

• \( a_1 = 1 \)
• \( a_2 = 1 \)
• \( a_3 = 1 \)

These values mean that when you're about to calculate a new term, there is no need to look further back beyond these initial specified terms. They are the anchors that help determine any subsequent value based on the recursive relationship established. Without these initial terms, the sequence remains undefined, as we always need a known point from which to start generating the next terms.

Sequence Calculation Steps

When calculating the terms of a recursively defined sequence, it is crucial to follow a set sequence of steps:1. **Start with Initial Terms**: Begin with the initial terms \( a_1, a_2, \) and \( a_3 \) provided in the exercise.2. **Apply Recursive Formula**: Use the recursive formula to find new terms. For example, to find \( a_4 \), use: \[ a_4 = a_3 + a_2 + a_1 = 1 + 1 + 1 = 3 \]3. **Continue to Next Terms**: Using the previously found terms, find the subsequent terms. For \( a_5 \), calculate as: \[ a_5 = a_4 + a_3 + a_2 = 3 + 1 + 1 = 5 \]4. **Repeat as Necessary**: Continue this process for as many terms as needed.This approach ensures that you stay organized and avoid confusion. By breaking down tasks into smaller, manageable pieces, calculations for recursive sequences become straightforward and easy to follow. It highlights the systematic nature of sequences where each term follows logically from its predecessors in a predictable pattern.