Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$3,7,11,15,19, \ldots$$

Step-by-Step Solution

Answer

The nth term of the sequence is \(4n - 1\).

Step 1: Identify the first term and common difference

The first term (a) of the sequence is 3 and the common difference (d), or the increment to the next term, is 4. That is, subtracting any term from the one following it results in 4.

Step 2: Apply the formula for the nth term of an arithmetic sequence

Apply the formula a + (n-1)d, where a is the first term and d is the common difference. Substitute the values from Step 1 into this formula to yield the expression for the nth term.

Step 3: Simplify the expression

Simplify the expression 3 + (n-1)4 to obtain the simplified form, 4n - 1. This is the nth term of the sequence.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a specific pattern, where each term after the first is obtained by adding a constant number, known as the common difference. This type of sequence follows a linear pattern and is one of the simplest forms of sequences in mathematics. Each term can be represented by an algebraic expression involving its position in the sequence, typically denoted by the variable 'n', which stands for the term number.

For example, in the sequence \(3, 7, 11, 15, 19, \ldots\), we start at 3 and add 4 to get to the next term, which is 7, and continue to add 4 to each term to move to the next. This constant addition creates a predictable pattern that can be capsulated into a general formula for finding any term in the sequence.

Common Difference

The common difference in an arithmetic sequence is the uniform amount by which consecutive terms increase or decrease. To find the common difference, you subtract any term from the subsequent term. This number is a key feature of the sequence and is denoted by the letter 'd'.

In the provided sequence \(3, 7, 11, 15, 19, \ldots\), finding the common difference involves a simple subtraction. If you take the second term (7) and subtract the first term (3), you'll get 4. This process is repeated for other consecutive terms, verifying that the common difference is indeed a constant value. Knowing the common difference is crucial when you need to determine the formula that expresses the nth term of an arithmetic sequence.

Sequence Patterns

Understanding sequence patterns is essential for identifying and describing the systematic structure within sequences. Sequence patterns determine the method for progressing from one term to the next and directly contribute to the way sequences are utilized in various mathematical applications.

With arithmetic sequences, the pattern is linear growth or decay by a fixed amount. This means that with each step from one term to the next, you consistently add or subtract the same value—the common difference. Recognizing these patterns not only allows one to predict future terms in the sequence but also to backtrack to earlier terms. Moreover, these patterns enable the formulation of an expression for the nth term, empowering you to calculate any term's value without listing the entire sequence.