Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$7,13,19,25,31, \ldots$$

Step-by-Step Solution

Answer

Expression for the \(n\)th term of the sequence is \(a_n = 7 + (n-1)*6\).

Step 1: Identify the pattern

Firstly, try to identify the pattern in the sequence. Here, it's noticed that for each subsequent term, 6 is added. So, the difference between each term (common difference) is 6.

Step 2: Get the first term

The first term (designated as \(a_1\)) in the sequence is \(7.\)

Step 3: Write the formula for \(n\)th term

The general formula for the \(n\)th term of an arithmetic sequence is \(a_n = a_1 + (n-1)*d\), where \(a_n\) is the \(n\)th term, \(a_1\) is the first term, and \(d\) is the common difference.

Step 4: Substituting the values into the formula

Substitute the values into the formula. For this sequence, \(a_1\) is \(7\) and the common difference, \(d\), is \(6\). Substituting these into the formula, we get \(a_n = 7 + (n-1)*6\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is referred to as the common difference. In our exercise, the sequence given is 7, 13, 19, 25, 31, and so on. When we examine this sequence, it becomes evident that each number is created by adding 6 to the previous number.

The power of an arithmetic sequence lies in its predictability and the ease with which one can determine any term in the series. This predictability is harnessed by a formula that defines how any term of the sequence can be found. In essence, if you know the first term and the common difference, you can easily compute the nth term, which is critical in solving problems related to such sequences.

Common Difference

The common difference is a key concept in understanding arithmetic sequences. It is the fixed amount that each term in the sequence increases or decreases by from its predecessor. In our problem, the common difference is 6, as it's the result of subtracting any term from the following term (for instance, 13 - 7 = 6 or 19 - 13 = 6).

Recognizing the common difference allows us to see the pattern in the sequence and is essential for writing the formula that will enable us to find the nth term. The formula incorporates the common difference as a multiplier that scales with the position of the term we're seeking to calculate, effectively tailoring the sequence to match the particular pattern we've identified.

Sequence Pattern

Understanding the sequence pattern is crucial in identifying the nature of any mathematical sequence. In an arithmetic sequence, the pattern is linear, showcasing a sequence where each term increases (or decreases) by a fixed amount—the common difference. By recognizing this linear pattern, we can articulate a rule for constructing the sequence.

To elaborate further on the sequence presented in the exercise, once the pattern is identified—which, in this case, is an increase by 6—we use this pattern to derive a general expression for the nth term. This application of the sequence pattern is fundamental to solving many mathematical problems and tasks, such as predicting future events or finding the value of a particular term in a sequence without having to list all the preceding ones.