Problem 33 - Precalculus: Mathematics for Calculus

Question

Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$4,10,16,22, \dots$$

Step-by-Step Solution

Answer

Common difference is 6; fifth term is 28; \(n\)th term is \(6n - 2\); 100th term is 598.

Step 1: Identify the Common Difference

In an arithmetic sequence, the common difference is the constant amount added to each term to get the next term. Subtract the first term from the second term: \(10 - 4 = 6\). This means the common difference, \(d\), is \(6\).

Step 2: Find the Fifth Term

To find the fifth term, use the formula for the \(n\)th term of an arithmetic sequence: \(a_n = a_1 + (n-1)\times d\). Here, \(a_1 = 4\), \(d = 6\), and \(n = 5\). So, \(a_5 = 4 + (5-1)\times 6 = 4 + 24 = 28\). The fifth term is \(28\).

Step 3: Find the General \(n\)th Term Formula

The general formula for the \(n\)th term \(a_n\) is given by \(a_n = a_1 + (n-1)\times d\). Substitute the values of \(a_1 = 4\) and \(d = 6\): \(a_n = 4 + (n-1) \times 6 = 4 + 6n - 6\). Simplify to get \(a_n = 6n - 2\).

Step 4: Calculate the 100th Term

Use the formula for the \(n\)th term, \(a_n = 6n - 2\), to find the 100th term. Set \(n = 100\): \(a_{100} = 6\times 100 - 2 = 600 - 2 = 598\). The 100th term is \(598\).

Key Concepts

Common Difference

In an arithmetic sequence, the "common difference" (\( d \)) is a crucial concept. It represents the constant amount that separates each term from the next in the sequence. To find it, simply subtract any term from the one that follows it. For instance, in the sequence provided: \( 4, 10, 16, 22, \dots \), we take \( 10 \) (the second term) and subtract the first term, \( 4 \).
Thus, \( d = 10 - 4 = 6 \). This tells us that each number is \( 6 \) units apart from the next.

• The common difference is consistent and applies to every step between terms.
• Ensure accuracy: always use consecutive terms to find \( d \)

Nth Term Formula

The "nth term formula" helps you determine any specific term in an arithmetic sequence without listing all the terms. This is highly useful when you're looking for a term far into the sequence, like the 100th term. The formula is:\[a_n = a_1 + (n-1) \times d\]where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term,
• \( n \) is the term number,
• and \( d \) is the common difference.

For example, to find the 5th term of the sequence \( 4, 10, 16, 22, \dots \), with \( a_1 = 4 \) and \( d = 6 \):\[a_5 = 4 + (5-1) \times 6 = 4 + 24 = 28\]

Arithmetic Sequence Formula

The "Arithmetic Sequence Formula" provides a structured method to describe all the terms of the sequence with a single formula known as the general term. Using the general formula: \( a_n = a_1 + (n-1) \times d \), substituting the known values \( a_1 = 4 \) and \( d = 6 \), gives:
\[a_n = 4 + (n-1) \times 6\]This simplifies to:\[a_n = 4 + 6n - 6\]Finally, it reduces to:\[a_n = 6n - 2\]This simple form allows you to find any term directly by substituting the term number for \( n \).

• Useful for finding distant terms like the 100th term.
• Makes predicting future terms straightforward without listing each one.

General Term in Sequences

The concept of the "General Term in Sequences" gives a comprehensive overview of the pattern in an arithmetic sequence. It is expressed with the formula previously mentioned:\( a_n = 6n - 2 \). This equation is your roadmap, helping to quickly find any term number \( n \) you desire. By substituting different values of \( n \), any term in the sequence can be computed. Consider finding the 100th term:\[a_{100} = 6 \times 100 - 2 = 598\]The significance of this formula lies in its ability to make sequences predictable and manageable:

• Allows you to calculate terms easily without constructing the sequence.
• Helps in recognizing the pattern or law governing the sequence.
• Ensures the sequence's consistency and logical structure.