Problem 47 - Algebra for College Students

Question

A fabric designer must take into account the capability of textile machines to produce material with vertical repeats. A textile machine can be set up for a vertical repeat every \(\frac{27}{n}\) inches (in.), where \(n\) is a natural number. Write the first five terms of the sequence \(a_{n}=\frac{27}{n},\) which gives the possible vertical repeats for a textile machine.

Step-by-Step Solution

Answer

27, 13.5, 9, 6.75, 5.4

Step 1: Identify the Sequence Formula

The given sequence formula is \(a_{n} = \frac{27}{n}\).

Step 2: Determine the First Term \(a_{1}\)

Substitute \(n = 1\) into the formula to get the first term: \(a_{1} = \frac{27}{1} = 27\).

Step 3: Determine the Second Term \(a_{2}\)

Substitute \(n = 2\) into the formula to get the second term: \(a_{2} = \frac{27}{2} = 13.5\).

Step 4: Determine the Third Term \(a_{3}\)

Substitute \(n = 3\) into the formula to get the third term: \(a_{3} = \frac{27}{3} = 9\).

Step 5: Determine the Fourth Term \(a_{4}\)

Substitute \(n = 4\) into the formula to get the fourth term: \(a_{4} = \frac{27}{4} = 6.75\).

Step 6: Determine the Fifth Term \(a_{5}\)

Substitute \(n = 5\) into the formula to get the fifth term: \(a_{5} = \frac{27}{5} = 5.4\).

Key Concepts

Natural Numbers

Natural numbers are a basic concept in mathematics. They include all the whole numbers starting from 1, 2, 3, and so on. They do not include zero or negative numbers. In the context of the given exercise, the variable \(n\) represents natural numbers since it only takes positive integer values. For example:

• 1
• 2
• 3
• 4
• 5

These values are then substituted into the sequence formula to get different terms. Understanding natural numbers is crucial because they form the backbone of many mathematical sequences and series.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. However, the given sequence is not arithmetic. Instead, it's a sequence where the terms are defined by the formula \(a_{n} = \frac{27}{n}\). Even though this is not a classic arithmetic sequence, it gives us a way to generate a sequence of numbers based on a clear mathematical rule. Each term depends on the previous term through division by a natural number.

Substitution

Substitution is a key mathematical technique used to solve many types of problems. It involves replacing a variable with its actual value to compute a specific term. In the given exercise, we substitute different natural numbers for \(n\) into the formula \(a_{n} = \frac{27}{n}\). Here's how it works:

• For the first term, substitute \(n = 1\): \(a_{1} = \frac{27}{1} = 27\).
• For the second term, substitute \(n = 2\): \(a_{2} = \frac{27}{2} = 13.5\).
• For the third term, substitute \(n = 3\): \(a_{3} = \frac{27}{3} = 9\).
• For the fourth term, substitute \(n = 4\): \(a_{4} = \frac{27}{4} = 6.75\).
• For the fifth term, substitute \(n = 5\): \(a_{5} = \frac{27}{5} = 5.4\).

By using substitution, you can calculate specific terms in a sequence and understand the pattern within the sequence.