Problem 24
Question
Define each recursively, where \(n \geq 0\) The \(n\) th term \(a_{n}\) of an arithmetic sequence with first term \(a\) and common difference \(d\)
Step-by-Step Solution
Verified Answer
The recursively defined \(n^{th}\) term \(a_n\) of an arithmetic sequence with first term \(a\) and common difference \(d\) is given by:
\[a_n =
\begin{cases}
a & \text{if } n = 0 \\
a_{n-1} + d & \text{if } n > 0
\end{cases}\]
1Step 1: Understand the properties of an arithmetic sequence
An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant, called the common difference (denoted by \(d\)). We are given the first term, \(a\), and we need to find a recursive formula for the rest of the terms in the sequence.
2Step 2: Define the base case
The base case of the sequence occurs when \(n = 0\). In this case, the term is the first term \(a\). Hence, we have
\[a_0 = a\]
3Step 3: Write the recursive formula in terms of the previous term
Since the difference between consecutive terms is constant, we can represent the \(n^{th}\) term of the sequence as a function of the previous term. We are looking for the term \(a_n\), where \(n>0\), which can be written as follows:
\[a_n = a_{n-1} + d\]
4Step 4: Combine the base case and recursive formula
Now we have the base case for \(n = 0\) and the recursive formula for \(n>0\). We can combine these together to create the complete definition of the sequence:
\[a_n =
\begin{cases}
a & \text{if } n = 0 \\
a_{n-1} + d & \text{if } n > 0
\end{cases}\]
Key Concepts
Recursive FormulaCommon DifferenceArithmetic Sequence PropertiesBase Case in Recursion
Recursive Formula
Understanding the recursive formula is key to grasping arithmetic sequences. A recursive formula allows us to find any term of a sequence by relating it to the terms before it. This is particularly useful when we don't need all previous terms, but just the immediately preceding one.
For an arithmetic sequence, the recursive formula incorporates a pattern of adding the common difference to the previous term. This looks like: \[a_n = a_{(n-1)} + d\]where \(a_n\)is the term you're trying to find, \(a_{(n-1)}\)is the term before it, and \(d\)is the common difference. An example would be a case where the first term is 3, the common difference is 5, and we're seeking the fourth term \(a_4\). We take the third term \(a_3\) and add 5 to find \(a_4\).
In educational content, it's essential to provide plenty of practice problems so learners can repeatedly apply this formula, which helps to reinforce the concept.
For an arithmetic sequence, the recursive formula incorporates a pattern of adding the common difference to the previous term. This looks like: \[a_n = a_{(n-1)} + d\]where \(a_n\)is the term you're trying to find, \(a_{(n-1)}\)is the term before it, and \(d\)is the common difference. An example would be a case where the first term is 3, the common difference is 5, and we're seeking the fourth term \(a_4\). We take the third term \(a_3\) and add 5 to find \(a_4\).
In educational content, it's essential to provide plenty of practice problems so learners can repeatedly apply this formula, which helps to reinforce the concept.
Common Difference
A non-negotiable characteristic of arithmetic sequences is the common difference, denoted by \(d\). It is the fixed amount added to each term to get the next term, and it is constant throughout the sequence. If the common difference is positive, the sequence increases; if it is negative, the sequence decreases.
For instance, in the sequence 4, 7, 10, 13, 16, the common difference is 3 because \(7 - 4 = 10 - 7 = 13 - 10 = 3\). Recognizing and calculating the common difference helps students understand the growth pattern of the sequence. It's essential to highlight exercises that ask students to identify or verify the common difference, because this skill sets the foundation for establishing the recursive formula.
For instance, in the sequence 4, 7, 10, 13, 16, the common difference is 3 because \(7 - 4 = 10 - 7 = 13 - 10 = 3\). Recognizing and calculating the common difference helps students understand the growth pattern of the sequence. It's essential to highlight exercises that ask students to identify or verify the common difference, because this skill sets the foundation for establishing the recursive formula.
Arithmetic Sequence Properties
The properties of arithmetic sequences are straightforward yet crucial for understanding these number patterns. Firstly, as we've highlighted, the sequence has a common difference. Another important property is that each term can be expressed as a combination of the first term and the common difference times the position of the term minus one. The general form is \[a_n = a_1 + (n-1)d\]where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
Other properties include the fact that the graph of an arithmetic sequence, with term number on the x-axis and term value on the y-axis, will be a straight line. Additionally, the sum of an arithmetic sequence can be found using the formula for the sum of n terms: \[S_n = \frac{n}{2}(2a_1 + (n-1)d)\]. Emphasizing these properties can help with various applications, such as finding the sum of a sequence or solving for unknown terms.
Other properties include the fact that the graph of an arithmetic sequence, with term number on the x-axis and term value on the y-axis, will be a straight line. Additionally, the sum of an arithmetic sequence can be found using the formula for the sum of n terms: \[S_n = \frac{n}{2}(2a_1 + (n-1)d)\]. Emphasizing these properties can help with various applications, such as finding the sum of a sequence or solving for unknown terms.
Base Case in Recursion
The concept of a base case is fundamental in recursion, especially within arithmetic sequences. The base case serves as the starting point for the sequence and prevents the recursion from continuing indefinitely. For an arithmetic sequence, the base case is often the first term, \(a_0\) or \(a_1\), depending on the context of the sequence.
In our example, the base case is given by \[a_0 = a\]. It provides the essential anchor from which all terms are calculated. When crafting educational content, it's important to stress the need for a base case when defining a sequence recursively, highlighting its value for preventing what is known as 'infinite recursion' or an endless loop. One useful exercise is to ask students to define various base cases and study how sequences evolve from different starting points.
In our example, the base case is given by \[a_0 = a\]. It provides the essential anchor from which all terms are calculated. When crafting educational content, it's important to stress the need for a base case when defining a sequence recursively, highlighting its value for preventing what is known as 'infinite recursion' or an endless loop. One useful exercise is to ask students to define various base cases and study how sequences evolve from different starting points.
Other exercises in this chapter
Problem 24
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