Question

The terms of a sequence are given recursively as \(a_{0}=2, a_{1}=6,\) and \(a_{n}=2 a_{n-1}+\) \(3 a_{n-2}\) for \(n \geq 2\). Find the first eight terms of this sequence.

Short answer

The sequence terms are 2, 6, 18, 54, 162, 486, 1458, 4374.

Step-by-step solution

Understanding the Problem

This is a recursively defined sequence. We have two initial terms, namely, \(a_0 = 2\) and \(a_1 = 6\), and a recurrence relation given by \(a_n = 2a_{n-1} + 3a_{n-2}\) for \(n \geq 2\). Our task is to find the first eight terms of this sequence.

Calculate Third Term

Using \(a_2 = 2a_{1} + 3a_{0} = 2(6) + 3(2)\), calculate the value: \(a_2 = 12 + 6 = 18\).

Calculate Fourth Term

Use the formula \(a_3 = 2a_{2} + 3a_{1}\): \(a_3 = 2(18) + 3(6)\), which simplifies to \(a_3 = 36 + 18 = 54\).

Calculate Fifth Term

For \(a_4\), compute: \(a_4 = 2a_{3} + 3a_{2} = 2(54) + 3(18) = 108 + 54 = 162\).

Calculate Sixth Term

Calculate \(a_5 = 2a_{4} + 3a_{3} = 2(162) + 3(54)\), which simplifies to \(a_5 = 324 + 162 = 486\).

Calculate Seventh Term

Determine \(a_6 = 2a_{5} + 3a_{4} = 2(486) + 3(162)\), leading to \(a_6 = 972 + 486 = 1458\).

Calculate Eighth Term

For \(a_7\), calculate: \(a_7 = 2a_{6} + 3a_{5} = 2(1458) + 3(486)\). This results in \(a_7 = 2916 + 1458 = 4374\).

Calculate Ninth Term

Find \(a_8 = 2a_{7} + 3a_{6} = 2(4374) + 3(1458)\), which gives \(a_8 = 8748 + 4374 = 13122\).

Key concepts

Recursive Sequence

A recursive sequence is a series of numbers where each term is defined as a function of one or more previous terms. This structure allows us to determine infinite terms of a sequence through an initial set of values and a specific formula or relation. In the given problem, the sequence is recursively defined, meaning new terms are calculated based on preceding terms using a predetermined rule.
Recursion provides a powerful way to model processes that are iterative, as it enables you to build complex patterns from simple, initial conditions. In our example, we start with two initial terms, and apply a formula: \(a_n = 2a_{n-1} + 3a_{n-2}\). This means each term is a combination of the two preceding terms, emphasizing the nature of recursive sequences.

• You begin with a basic seed value or values, termed as initial conditions.
• You use a recurrence relation to derive subsequent terms.

The elegance of recursive sequences lies in their ability to generate large datasets with minimal starting information and descriptive rules.

Initial Conditions

Initial conditions are the building blocks of a recursive sequence. They serve as the starting point from which the rest of the sequence is generated. Without initial conditions, a recursive sequence cannot begin.
In this problem, the initial conditions are given as \(a_0 = 2\) and \(a_1 = 6\). These terms provide us the necessary foundation to apply our recurrence relation and find further terms in the sequence.
The idea is simple:

• Define the first few terms explicitly. These terms are chosen based on the problem's requirements or constraints.
• Use them as starting points to employ the recurrence relation and compute further terms as needed.

With these starting values, we can generate the subsequent numbers of the sequence, ensuring every term correctly follows the established rule based on its predecessors.

Discrete Mathematics

Discrete mathematics deals with structures that are distinct and separate, rather than continuous. This field includes topics such as logic, set theory, and number theory. One important aspect of discrete mathematics is the study of sequences, particularly recursive sequences, which are subsets of integers or whole numbers predicted through defined formulas.
In our exercise, we leverage discrete mathematics principles to analyze and predict terms in a sequence using the given recurrence relation.
Here's why discrete mathematics is important in studying sequences:

• It provides a foundation for understanding how sequences behave over different terms.
• It allows for systematic calculations and predictions, crucial for sequences that extend to infinite terms.

By applying discrete mathematics, we ensure the consistency and logical progression of terms in our sequence, following the rules and initial conditions precisely.

Step-by-Step Solution

A step-by-step solution helps break down complex problems into manageable pieces. It involves solving problems one step at a time, ensuring each part is understood before moving to the next. This method is especially helpful in recursive sequences where each term depends on those calculated before it.
Our solution illustrates this clearly:

• We start by identifying the problem and the goal: find the first eight terms.
• Next, we use the initial conditions and the recurrence relation to compute each term systematically, starting with \(a_2\) and moving up to \(a_8\).
• Each step expands on the previous one, applying the rule \(a_n = 2a_{n-1} + 3a_{n-2}\), ensuring accuracy and clarity.

This step-by-step progression not only guides you through the solution but also ensures that all parts of the problem are addressed, and each term is correctly calculated, building a comprehensive understanding of recursive sequences.