Problem 28 - Precalculus: Mathematical for Calculus

Question

Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$\frac{7}{6}, \frac{5}{3}, \frac{13}{6}, \frac{8}{3}, \dots$$

Step-by-Step Solution

Answer

The common difference is \( \frac{1}{2} \), fifth term is \( \frac{19}{6} \), \( n \)th term is \( \frac{3n+4}{6} \), and 100th term is \( \frac{152}{3} \).

Step 1: Identify the Common Difference

An arithmetic sequence is defined by the constant difference between consecutive terms. To find this 'common difference', subtract the first term from the second term.The first term is \( \frac{7}{6} \) and the second term is \( \frac{5}{3} \). Convert \( \frac{5}{3} \) to have a common denominator with \( \frac{7}{6} \):\[ \frac{5}{3} = \frac{10}{6} \]Then subtract the first term:\[ \frac{10}{6} - \frac{7}{6} = \frac{3}{6} = \frac{1}{2} \]Thus, the common difference is \( \frac{1}{2} \).

Step 2: Calculate the Fifth Term

The fifth term can be found using the formula for the \( n \)th term in an arithmetic sequence, which is \( a_n = a_1 + (n-1) \times d \), where \( a_1 \) is the first term and \( d \) is the common difference.Here, \( a_1 = \frac{7}{6} \), \( d = \frac{1}{2} \), and \( n = 5 \).Calculate the fifth term:\[ a_5 = \frac{7}{6} + (5-1) \times \frac{1}{2} \]\[ a_5 = \frac{7}{6} + 4 \times \frac{1}{2} \]\[ a_5 = \frac{7}{6} + 2 = \frac{7}{6} + \frac{12}{6} = \frac{19}{6} \]So, the fifth term is \( \frac{19}{6} \).

Step 3: Determine the nth Term

The formula for the \( n \)th term in an arithmetic sequence is \( a_n = a_1 + (n-1) \times d \).Substitute \( a_1 = \frac{7}{6} \) and \( d = \frac{1}{2} \) into the formula to find the general expression for \( a_n \):\[ a_n = \frac{7}{6} + (n-1) \times \frac{1}{2} \]Simplify:\[ a_n = \frac{7}{6} + \frac{n-1}{2} \]To simplify further, express \( \frac{n-1}{2} \) in terms of sixths:\[ \frac{n-1}{2} = \frac{3(n-1)}{6} = \frac{3n-3}{6} \]Combine the terms:\[ a_n = \frac{7}{6} + \frac{3n-3}{6} = \frac{3n + 4}{6} \]

Step 4: Calculate the 100th Term

Use the formula derived in the previous step to find the 100th term. Substitute \( n = 100 \) into the formula \( a_n = \frac{3n + 4}{6} \):\[ a_{100} = \frac{3(100) + 4}{6} \]\[ a_{100} = \frac{300 + 4}{6} = \frac{304}{6} \]Simplify \( \frac{304}{6} \):\[ a_{100} = \frac{304}{6} = \frac{152}{3} \]Therefore, the 100th term is \( \frac{152}{3} \).

Key Concepts

Understanding the Common Difference

The backbone of an arithmetic sequence is the concept of the 'common difference.' This is the fixed amount that you add to get from one term to the next. In our example sequence, to find the common difference, we subtract the first term from the second term. The sequence begins with \( \frac{7}{6} \) and the next term is \( \frac{5}{3} \). Before diving into subtraction, it's important that both fractions share a common denominator. So, we convert \( \frac{5}{3} \) to \( \frac{10}{6} \).
Next, we subtract \( \frac{7}{6} \) from \( \frac{10}{6} \):

• \( \frac{10}{6} - \frac{7}{6} = \frac{3}{6} = \frac{1}{2} \)

This tells us that the common difference, often denoted by \( d \), is \( \frac{1}{2} \). This means each term is \( \frac{1}{2} \) greater than the one before it, making calculations simple once identified.

Exploring the nth Term Formula

In arithmetic sequences, the formula for the nth term helps us find any term we seek without listing all previous terms. The formula is:
\[ a_n = a_1 + (n-1) \times d \]
Let's break it down:

• \( a_n \) represents the term at position \( n \).
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference.

To find a specific term, such as the fifth term:
- Here, \( a_1 = \frac{7}{6} \) and \( d = \frac{1}{2} \). For the fifth term (\( n = 5 \)):
\[ a_5 = \frac{7}{6} + (5-1) \times \frac{1}{2} = \frac{19}{6} \]
For a general term, we simplify:
\[ a_n = \frac{7}{6} + \frac{n-1}{2} = \frac{3n + 4}{6} \]
This formula, \( \frac{3n + 4}{6} \), is a compact expression to find any term in our sequence.

Arithmetic Series Calculation: Finding Specific Terms

Sometimes, we seek to find not just the next term but perhaps the 100th or any other high-numbered term in a sequence. Instead of counting one by one, we use the simplified nth term formula.
Previously, we derived that formula as \( a_n = \frac{3n + 4}{6} \).
To find the 100th term, simply set \( n = 100 \) in the formula:
\[ a_{100} = \frac{3(100) + 4}{6} = \frac{304}{6} \]
Simplify \( \frac{304}{6} \) to get \( \frac{152}{3} \).
This approach not only allows you to quickly find the 100th term but any term you need, illustrating the power of the nth term formula in efficiently handling arithmetic sequences.