Problem 15 - Discrete Mathematics with Applications

Question

Define recursively each sequence of numbers. (Hint: Look for a pattern and define the \(n\) th term \(a_{n}\) recursively.) $$1,4,7,10,13 \ldots$$

Step-by-Step Solution

Answer

The sequence can be defined recursively as: $$ a_n = \begin{cases} 1, & \text{if } n = 1 \\ a_{n-1} + 3, & \text{if } n > 1 \end{cases} $$

Step 1: Identify the first term

Start by identifying the first term of the sequence, which is the base case. Here, the first term is \(a_1 = 1\).

Step 2: Determine the pattern

Now, to identify the pattern or the relationship between consecutive terms, we can look at the differences between consecutive terms: \(4 - 1 = 3\) \(7 - 4 = 3\) \(10 - 7 = 3\) \(13 - 10 = 3\) From this, we can see that each term can be obtained by adding 3 to the previous term.

Step 3: Define the recursive formula

To define the sequence recursively, we can use the pattern identified in Step 2. For any term \(a_n\), for \(n > 1\), we have: $$ a_n = a_{n-1} + 3 $$

Step 4: Write the final recursive definition

Now that we have the recursive formula and the base case, we can combine them to provide a complete recursive definition for the sequence: $$ a_n = \begin{cases} 1, & \text{if } n = 1 \\ a_{n-1} + 3, & \text{if } n > 1 \end{cases} $$ This recursive definition captures the given sequence of numbers and can be used to find any term in the sequence.

Key Concepts

Recursively Define Sequence

In mathematics, sequences are often described with a formula that expresses each term using one or more of its predecessors. To recursively define a sequence means to specify the first term (or few terms), known as the base case, and then define each subsequent term based on previous terms.

For example, in the given sequence \(1,4,7,10,13 \ldots\), the recursive definition involves two components: one stating that the first term \(a_1=1\), and the second describing how to obtain any term \(a_n\) when \(n>1\) by adding 3 to the preceding term \(a_{n-1}\). Thus, we have:
\[ a_n = \begin{cases}1, & \text{if } n = 1 \a_{n-1} + 3, & \text{if } n > 1\end{cases} \]
The beauty of recursive definitions lies in their simplicity and their powerful application in computational processes where the algorithm can simply use the previous outcomes to generate new ones.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference, called the common difference, to the previous term. This sequence forms a linear pattern and is one of the simplest types of number patterns in mathematics.

To spot an arithmetic sequence, take the sequence \(1,4,7,10,13 \ldots\) as an example. The difference between consecutive terms is always the same:\[4 - 1 = 3\]\[7 - 4 = 3\]\[10 - 7 = 3\]\[13 - 10 = 3\]
This repetitive addition of 3 is the hallmark of an arithmetic sequence with 3 being the common difference. We can express the \(n\)th term of an arithmetic sequence with the formula: \(a_n = a_1 + (n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = 1\) and \(d = 3\), which can then be used to explicitly determine any term in the sequence.

Mathematical Induction

Mathematical induction is a proof technique used to establish the validity of a property for all natural numbers. It involves two crucial steps: showing the base case is true, and then proving that if it's true for one case, it must be true for the next.

To apply mathematical induction to an arithmetic sequence like \(1,4,7,10,13 \ldots\), for the base case, we'd verify that the formula for the \(n\)th term correctly computes the first term. Then, we assume that the formula holds for some arbitrary term \(n\) and prove that it will also hold for term \(n+1\). This step typically involves algebraic manipulation and substitution into the assumed formula for term \(n\), then showing it leads to a correct expression for term \(n+1\).

While the exercise provided deals with a specific sequence, mathematical induction can be utilized as a powerful proof tool in various contexts, such as inequalities, divisibility, and other properties of numbers.