Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the sequence. \(a_{1}=9, \quad a_{n}=a_{n-1}+n\)

Step-by-Step Solution

Answer

The first five terms are 9, 11, 14, 18, 23.

Step 1: Understand the Base Term

The given sequence starts from the first term, which is explicitly given to us: \(a_1 = 9\). This is the initial term that will be used to generate the subsequent terms of the sequence.

Step 2: Apply the Recursive Formula for the Second Term

The next term of the sequence is defined recursively by the formula \(a_n = a_{n-1} + n\). For the second term, \(a_2 = a_1 + 2 = 9 + 2 = 11\).

Step 3: Calculate the Third Term

Using the recursive formula \(a_n = a_{n-1} + n\), we find the third term: \(a_3 = a_2 + 3 = 11 + 3 = 14\).

Step 4: Determine the Fourth Term

Continuing with the same formula \(a_n = a_{n-1} + n\), the fourth term is \(a_4 = a_3 + 4 = 14 + 4 = 18\).

Step 5: Find the Fifth Term

Finally, use the recursive formula to calculate the fifth term: \(a_5 = a_4 + 5 = 18 + 5 = 23\).

Key Concepts

Sequence Terms

A sequence is an ordered list of numbers, called terms, that follow a specific pattern or rule. Each number in the sequence is a term and is typically designated by a subscript index to show its position. In our case, we are asked to write the first five terms of a sequence where the terms are defined recursively.

For example, the first term, often denoted as \(a_1\), provides the starting point. Here, it equals 9. This is the base term provided in the problem. The sequence demands applying a recursive rule to find subsequent terms based on the previous one.

Having clearly defined sequence terms is crucial because they build a foundation to establish patterns or relationships among them. This pattern can then be applied to find any term within the sequence without listing everything before it. Understanding these can help reveal intricate structures or characteristics inherent in sequences.

Recursive Formula

A recursive formula allows us to calculate the terms in a sequence using the preceding term or terms. In essence, it states how to transform the prior term into the next one, introducing a sense of progression. The formula in this exercise, \(a_n = a_{n-1} + n\), indicates we should add the term number \(n\) to its immediate predecessor \(a_{n-1}\) to get the term \(a_n\).

To use a recursive formula effectively:

• Start with a given initial term (if provided).
• Apply the recursive rule to determine the next term.
• Repeat the process to find subsequent terms.

Knowing how a recursive formula functions is key to understanding how the terms relate to each other and calculate them sequentially. It's an essential tool in algebra when dealing with sequences, providing a systematic approach to predict future elements.

Algebraic Sequences

Algebraic sequences are sequences defined by a specific algebraic rule. This can be an explicit formula or, as in our exercise, a recursive one. These rules employ algebraic operations like addition, subtraction, multiplication, or division.

Such sequences exhibit numerical consistency, allowing them to be analyzed through the lens of algebra. Whether through recursive or explicit representations, algebraic sequences can represent a wide variety of models in both mathematics and real-world applications.

In the exercise, the sequence is given by a recursive rule \(a_n = a_{n-1} + n\). Here we see how a simple arithmetic operation—adding the term's position—defines the sequence. Recognizing a sequence as algebraic helps in predicting its future behavior.