Problem 29 - Algebra and Trigonometry

Question

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 4,9,14,19, \dots $$

Step-by-Step Solution

Answer

The common difference is 5, the fifth term is 24, the nth term is \(4 + (n-1) imes 5\), and the 100th term is 499.

Step 1: Identify the First Term

The first term of the sequence, denoted as \( a_1 \), is the first number given in the sequence, which is 4.

Step 2: Determine the Common Difference

The common difference \( d \) in an arithmetic sequence is found by subtracting any term from the subsequent term. Here, \( d = 9 - 4 = 5 \).

Step 3: Find the Fifth Term

To compute the fifth term, \( a_5 \), use the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n-1) imes d \). Substitute \( n = 5 \), \( a_1 = 4 \), and \( d = 5 \): \( a_5 = 4 + (5-1) imes 5 = 24 \).

Step 4: Derive the General Formula for the nth Term

The formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1) imes d \). With \( a_1 = 4 \) and \( d = 5 \), the nth term formula becomes \( a_n = 4 + (n-1) imes 5 \).

Step 5: Calculate the 100th Term

Using the formula for the nth term, calculate the 100th term: \( a_{100} = 4 + (100-1) imes 5 = 499 \).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key element. It describes the difference between consecutive terms in the sequence. Understanding it helps to predict subsequent terms before expanding the sequence. The common difference, denoted as \(d\), is consistent throughout an arithmetic sequence.

• To find \(d\), subtract any term from the one that follows. For example, in our sequence \(4, 9, 14, 19,\dots\), \(d = 9 - 4 = 5\).
• This consistency makes arithmetic sequences predictable and easy to work with.

By knowing the common difference, you're well on your way to identifying the pattern of the sequence.

nth Term Formula

The nth term formula gives a general expression for finding any term in an arithmetic sequence. This is particularly useful when you need to find terms that are far apart. The formula is expressed as \( a_n = a_1 + (n-1) \times d \), where:

• \(a_n\) represents the nth term you're trying to find.
• \(a_1\) is the first term of the sequence.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference.

In our sequence, with \(a_1 = 4\) and \(d = 5\), the nth term formula becomes \( a_n = 4 + (n-1) \times 5 \). This formula simplifies finding any term in the sequence without listing all previous terms.

Fifth Term

Finding the fifth term in an arithmetic sequence is simple with the nth term formula. Plugging in the respective values makes the calculation straightforward. For our example sequence, the fifth term, \(a_5\), is calculated as follows:

• Use the nth term formula: \( a_n = a_1 + (n-1) \times d \).
• Set \(n = 5\), \(a_1 = 4\), and \(d = 5\).
• Calculate: \( a_5 = 4 + (5-1) \times 5 = 24 \).

The fifth term, \(24\), fits perfectly into the pattern established by our common difference. This demonstrates how effective the nth term formula is in locating any term of interest.

100th Term

Calculating the 100th term of an arithmetic sequence might seem daunting at first. However, with the nth term formula, it's as simple as replacing a few variables. Here's how to find the 100th term in our example sequence:

• Use the nth term formula: \(a_n = a_1 + (n-1) \times d\).
• Input \(n = 100\), \(a_1 = 4\), \(d = 5\).
• Compute: \(a_{100} = 4 + (100-1) \times 5 = 499\).

The result, 499, shows the power of the formula in expanding an arithmetic sequence to find terms far along in the series. It’s a clear way to project values without manual listing.