Problem 18
Question
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
Verified Answer
The first five terms are 1, 1, 1, 3, and 5.
1Step 1: Understand the Problem
We are given a recursively defined sequence where each term starting from the fourth is the sum of the previous three terms. The initial terms are provided: \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\). We need to find the first five terms of the sequence.
2Step 2: List the Initial Terms
We already know the values for the first three terms of the sequence: \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\). These values will be used to calculate subsequent terms.
3Step 3: Calculate the Fourth Term
Using the recursive formula \(a_n = a_{n-1} + a_{n-2} + a_{n-3}\), calculate the fourth term: \(a_4 = a_3 + a_2 + a_1 = 1 + 1 + 1 = 3\).
4Step 4: Calculate the Fifth Term
Now, use the recursive formula to find the fifth term: \(a_5 = a_4 + a_3 + a_2 = 3 + 1 + 1 = 5\).
5Step 5: Compile the Results
We now have 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
Sequence TermsRecursive DefinitionInitial Conditions
Sequence Terms
In mathematics, a sequence is simply a list of numbers that follow a certain rule. Each number in this list is known as a term.
When we talk about sequence terms, we are essentially discussing the individual numbers that make up the sequence.
Each term can be referred to by its position in the sequence. For example, the first term is called \(a_1\), the second term is \(a_2\), and so on. In the case of our recursively defined sequence, the terms are determined using the values that come before them.
The initial terms of this specific sequence are known as \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\). These terms are very important, as they form the basis from which all subsequent terms are calculated.
Understanding this concept is crucial as it helps you to identify patterns and predict future terms in a sequence.
When we talk about sequence terms, we are essentially discussing the individual numbers that make up the sequence.
Each term can be referred to by its position in the sequence. For example, the first term is called \(a_1\), the second term is \(a_2\), and so on. In the case of our recursively defined sequence, the terms are determined using the values that come before them.
The initial terms of this specific sequence are known as \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\). These terms are very important, as they form the basis from which all subsequent terms are calculated.
Understanding this concept is crucial as it helps you to identify patterns and predict future terms in a sequence.
Recursive Definition
A recursive definition defines something in terms of itself. For sequences, this means that each term is defined using previous terms.
The given exercise involves a recursive sequence where each term from the fourth onward is calculated as the sum of the preceding three terms.
In formulaic terms, the recursive definition provided is \(a_n = a_{n-1} + a_{n-2} + a_{n-3}\).
This means that to find a term, you simply add up the three terms that came before it.
This definition is like a set of instructions that tells you how to calculate each term after the initial terms.
Such definitions are powerful because they provide a compact way to define complex sequences with only a few statements.
The given exercise involves a recursive sequence where each term from the fourth onward is calculated as the sum of the preceding three terms.
In formulaic terms, the recursive definition provided is \(a_n = a_{n-1} + a_{n-2} + a_{n-3}\).
This means that to find a term, you simply add up the three terms that came before it.
This definition is like a set of instructions that tells you how to calculate each term after the initial terms.
Such definitions are powerful because they provide a compact way to define complex sequences with only a few statements.
Initial Conditions
Initial conditions for a sequence are the starting points from which the sequence is generated.
For recursive sequences, these are essential because they give us the first few terms needed to start the pattern described by the recursive definition.
In the exercise, the initial conditions are given as \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\).
Without these initial conditions, we would not have enough information to calculate any new terms in the sequence.
They work as the foundational building blocks for the entire sequence.
Think of them as the seed that grows into the entire sequence tree, offering us endless possibilities to explore as further terms are calculated.
For recursive sequences, these are essential because they give us the first few terms needed to start the pattern described by the recursive definition.
In the exercise, the initial conditions are given as \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\).
Without these initial conditions, we would not have enough information to calculate any new terms in the sequence.
They work as the foundational building blocks for the entire sequence.
Think of them as the seed that grows into the entire sequence tree, offering us endless possibilities to explore as further terms are calculated.
Other exercises in this chapter
Problem 18
Show that \(n^{3}-n+3\) is divisible by 3 for all natural numbers \(n\)
View solution Problem 18
Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$\ln 2, \ln 4, \ln 8, \ln 16, \dots$$
View solution Problem 19
Dr. Gupta is considering a 30 -year mortgage at \(6 \%\) interest. She can make payments of \(\$ 3500\) a month. What size loan can she afford?
View solution Problem 19
Determine whether the sequence is geometric. If it is geometric, find the common ratio. $$1.0,1.1,1.21,1.331, \ldots$$
View solution