Problem 33 - 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. $$ 25,26.5,28,29.5, \dots $$

Step-by-Step Solution

Answer

The common difference is 1.5, the fifth term is 31, the general formula is \( a_n = 1.5n + 23.5 \), and the 100th term is 173.5.

Step 1: Identify the First Term

The first term of the arithmetic sequence is given as 25. We denote this as \( a_1 = 25 \).

Step 2: Determine the Common Difference

To find the common difference \( d \), subtract the first term from the second term: \( d = 26.5 - 25 = 1.5 \).

Step 3: Calculate the Fifth Term

The \( n \)-th term of an arithmetic sequence is given by the formula \( a_n = a_1 + (n - 1)d \). For the fifth term \( a_5 \), substitute \( n = 5 \), \( a_1 = 25 \), and \( d = 1.5 \) into the formula: \( a_5 = 25 + (5 - 1) imes 1.5 = 25 + 6 = 31 \). Thus, the fifth term is 31.

Step 4: Derive the General Formula for the n-th Term

Using the formula \( a_n = a_1 + (n - 1)d \) and substituting \( a_1 = 25 \) and \( d = 1.5 \), the general term \( a_n \) can be written as \( a_n = 25 + (n - 1) imes 1.5 = 25 + 1.5n - 1.5 = 1.5n + 23.5 \).

Step 5: Calculate the 100th Term

Using the derived formula \( a_n = 1.5n + 23.5 \), substitute \( n = 100 \) to find the 100th term: \( a_{100} = 1.5 imes 100 + 23.5 = 150 + 23.5 = 173.5 \). Thus, the 100th term is 173.5.

Key Concepts

Understanding Common Difference

In an arithmetic sequence, the common difference is a key component that determines the uniformity of the sequence. It is the amount by which each term increases or decreases to form the next term. This constant difference is denoted by the letter \( d \). For example, in the sequence you provided: 25, 26.5, 28, 29.5, ..., the common difference can easily be found by subtracting the first term from the second term. This gives you:

• From 25 to 26.5, the difference is 1.5.

All terms follow this consistent increment of 1.5. All arithmetic sequences have this feature where \( d \) remains the same between any two consecutive terms.

Nth Term Formula in Arithmetic Sequences

The nth term formula is crucial for finding any term within an arithmetic sequence without listing all previous terms. This formula is given as:

• \( a_n = a_1 + (n-1) \cdot d \)

Here:

• \( a_n \) is the term you're looking for.
• \( a_1 \) is the first term in the sequence.
• \( n \) is the position of the term.
• \( d \) is the common difference.

By plugging the values into this formula, you can efficiently find any specific term, like the 5th or the 100th, saving time and effort.

General Term

The general term of an arithmetic sequence, symbolized as \( a_n \), provides a method for calculating any term within the sequence mathematically. This formula sets a pattern based on the sequence's first term and its common difference:

• For the sequence 25, 26.5, 28, ..., the general term is derived as follows:

Begin with our nth term formula: \( a_n = a_1 + (n-1) \cdot d \). Substitute the known values here:

• First term \( a_1 = 25 \)
• Common difference \( d = 1.5 \)

The formula becomes:

• \( a_n = 25 + (n-1) \cdot 1.5 = 1.5n + 23.5 \)

This general term equation lets you find any sequence term by just substituting the position \( n \), making it incredibly useful for calculations.

Sequence Terms

A series of numbers in a specific order form what is known as sequence terms. In an arithmetic sequence, these terms are arranged such that the difference between any two consecutive terms is constant. Understanding this concept allows you to generate and analyze sequences easily.In the given sequence starting with 25, the sequence terms are as follows based on the common difference:

• First term (\(a_1\)): 25
• Second term (\(a_2\)): 26.5
• Third term (\(a_3\)): 28

You continue this pattern using the common difference \(d = 1.5\) to find each next term. This method helps in visualizing the pattern and understanding how sequences grow.