Question

\(a_{n}=2 a_{n-1}+3 a_{n-2}\) for \(n \geq 2\) where \(a_{0}=2\) and \(a_{1}=2\).

Short answer

The sequence starts with 2, 2, 10, 26, 82.

Step-by-step solution

Understand the Problem

We are given a recurrence relation: \( a_n = 2a_{n-1} + 3a_{n-2} \), with specific initial conditions: \( a_0 = 2 \) and \( a_1 = 2 \). This means we need to find subsequent terms using these initial values and the given formula.

Calculate the Second Term

Using the recurrence relation, substitute \( n = 2 \): \( a_2 = 2a_1 + 3a_0 = 2(2) + 3(2) = 4 + 6 = 10 \). Thus, \( a_2 = 10 \).

Calculate the Third Term

Now calculate \( a_3 \) using the recurrence relation: \( a_3 = 2a_2 + 3a_1 = 2(10) + 3(2) = 20 + 6 = 26 \). Thus, \( a_3 = 26 \).

Calculate the Fourth Term

Calculate \( a_4 \) using the relation: \( a_4 = 2a_3 + 3a_2 = 2(26) + 3(10) = 52 + 30 = 82 \). Thus, \( a_4 = 82 \).

Key concepts

Understanding Initial Conditions

Initial conditions are the foundation from which any sequence begins its journey. In the context of recurrence relations, they act like starting points. They give us the very first values necessary to generate the rest of the sequence.
For the exercise, the initial conditions provided are:

• \( a_0 = 2 \)
• \( a_1 = 2 \)

These values are crucial because we start from them to calculate the rest of the terms using the given recurrence relation. Without initial conditions, it would be impossible to calculate any further terms, as there would be no base case to work from.
Understanding and correctly applying initial conditions ensures that the sequence generated is accurate and meaningful, aligning correctly with the recurrence relationship.

The Role of Discrete Mathematics

Discrete mathematics is an essential area of mathematics that deals with countable, distinct elements. It includes the study of objects such as integers, graphs, and logical statements. Recurrence relations, like the one given in this problem, are a part of discrete mathematics.
What sets discrete mathematics apart is its focus on sequences, sets, and logic, which are all distinct and do not require the notion of continuity found in calculus. This means that solutions like recurrence relations are boiled down to whole numbers or specific steps.
Recurrence relations, in particular, show how one element of a sequence is derived from others, step by step. In essence, discrete mathematics allows us to rigorously define processes involving anything that's countable. For students, it presents clear steps for reaching solutions and is invaluable for computer science, logic, and cryptography applications.

Calculating Sequence Terms

Sequence calculation is essentially the art of using a known formula to find the terms of a sequence. In our example, this process involves using the specified recurrence relation repeatedly, substituting in each preceding term.
The recurrence relation given:\[ a_{n} = 2a_{n-1} + 3a_{n-2} \]tells us how each term is formed from the two preceding terms.
The process involves repetitive substitution:

• For \( n = 2 \), substitute \( a_1 \) and \( a_0 \) to find \( a_2 \)
• For \( n = 3 \), use \( a_2 \) and \( a_1 \) to get \( a_3 \)
• Continue this way to find further terms

This method not only ensures the accuracy of each term but also confirms that the sequence aligns with the conditions given at the start. Understanding sequence calculation solidifies the student's comprehension of generating series, reinforcing concepts in arithmetic and geometric progressions as well. This approach is not only systematic but crucial for solving advanced problems in mathematics and computer science.