Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence defined recursively. $$a_{1}=15, a_{k}=a_{k-1}+3$$

Step-by-Step Solution

Answer

The first five terms of the sequence are 15, 18, 21, 24, 27

Step 1: Identify the first term

The first term, \(a_1\), is given as 15

Step 2: Apply the recursive rule to find the second term

In a recursive sequence, each term is based on the one before it. Here, we are told that each term, \(a_k\), is the previous term, \(a_{k-1}\), plus 3. Therefore, the second term, \(a_2\), is \(a_1\) plus 3, which equals 15 + 3 = 18.

Step 3: Repeat the recursive rule for the third, fourth and fifth terms

We just continue adding 3 to the previous term to get the next term. So, the third term, \(a_3\), is \(a_2\) plus 3, which equals 18 + 3 = 21. The fourth term, \(a_4\), is \(a_3\) plus 3, which equals 21 + 3 = 24. Finally, the fifth term, \(a_5\), is \(a_4\) plus 3, which equals 24 + 3 = 27.

Key Concepts

Arithmetic Sequences

In the realm of mathematics, an arithmetic sequence is a collection of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. An arithmetic sequence can be visualized as a simple numerical pattern where, starting from a particular number, each subsequent number is obtained by adding the common difference.

For example, if we begin with 2 and the common difference is 3, the arithmetic sequence would be 2, 5, 8, 11, and so forth. The structure of an arithmetic sequence is foundational for understanding more complex mathematical concepts and is often one of the first patterns students learn in sequence and series analysis.

Sequence Notation

Understanding sequence notation is crucial for clearly communicating mathematical ideas. Sequences are often denoted by an alphabetical character, commonly 'a', followed by a subscript that represents the position of a term within the sequence. For instance, in the notation \(a_n\), \(n\) signifies the nth term of the sequence.

Specific terms are referred to by substituting the position number for \(n\). Thus, \(a_1\) is the first term, \(a_2\) is the second term, and so on. This notation helps to establish a framework for articulating and analyzing the properties of sequences without ambiguity, making it easier to formulate and solve problems related to sequences.

Recursive Formula

A recursive formula defines each term of a sequence using the preceding terms, allowing for the sequential building of the sequence. It uniquely specifies how to get from one term to the next and differs from an explicit formula, which defines any term directly without reference to previous terms.

To illustrate, considering the recursive formula \(a_k = a_{k-1} + 3\) with a starting term of \(a_1 = 15\), one calculates each term by adding 3 to the term before it. The defining characteristic of a recursive formula is that knowledge of the preceding term is necessary to find the value of the current term. It is a fundamental concept for generating sequences and is also pivotal in computer science and algorithm design, where recursive processes frequently occur.