Problem 7 - A Graphical Approach to College Algebra

Question

Write the first five terms of each arithmetic sequence. Do not use a calculator. The first term is \(8,\) and the common difference is 6

Step-by-Step Solution

Answer

The first five terms are 8, 14, 20, 26, and 32.

Step 1: Understand the Problem

An arithmetic sequence is a sequence of numbers in which each term after the first is the sum of the previous term and a constant, known as the common difference. Here, the first term is given as 8, and the common difference is 6.

Step 2: Write the First Term

The first term of the sequence is already provided: \( a_1 = 8 \). This is the starting point of our sequence.

Step 3: Calculate the Second Term

To find the second term, add the common difference to the first term: \( a_2 = a_1 + 6 = 8 + 6 = 14 \).

Step 4: Calculate the Third Term

Add the common difference to the second term to find the third term: \( a_3 = a_2 + 6 = 14 + 6 = 20 \).

Step 5: Calculate the Fourth Term

Add the common difference to the third term to find the fourth term: \( a_4 = a_3 + 6 = 20 + 6 = 26 \).

Step 6: Calculate the Fifth Term

Finally, add the common difference to the fourth term to find the fifth term: \( a_5 = a_4 + 6 = 26 + 6 = 32 \).

Key Concepts

Arithmetic Sequence Formula

The arithmetic sequence formula is key to understanding how sequences work. In an arithmetic sequence, every term after the first is formed by adding a constant known as the 'common difference' to the preceding term. This pattern of adding the same amount each time distinguishes arithmetic sequences from other types of sequences.
The general formula for the nth term of an arithmetic sequence is given by:

• \( a_n = a_1 + (n-1) \cdot d \)

Here, \( a_n \) represents the nth term of the sequence, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
Using this formula, you can easily find any term in the sequence without needing to calculate all the previous terms. This formula is particularly useful for sequences with large numbers of terms.

Common Difference

The common difference is a crucial part of arithmetic sequences. It is the consistent amount added to each term to get the next term. Understanding the role of the common difference helps us identify and form arithmetic sequences.

• In our example, the sequence starts at 8 and the common difference is 6.

To find the next term in the sequence, simply add the common difference to the previous term. For instance, starting with 8, you add 6 to get 14, then another 6 to get 20, and so on.
The common difference can be positive, negative, or zero, leading to increasing, decreasing, or constant sequences respectively. It ultimately determines the direction and rate of change in the sequence.

Terms of a Sequence

The terms of an arithmetic sequence are the specific numbers generated by repeatedly adding the common difference to the first term. Each number or 'term' follows neatly on from the last in this linear sequence.
In our original problem, we started with the first term, which was given as 8. Each subsequent term was found by adding the common difference of 6:

• The first term \( a_1 \) is 8.
• The second term \( a_2 \) is 14 (8+6).
• The third term \( a_3 \) is 20 (14+6).
• The fourth term \( a_4 \) is 26 (20+6).
• The fifth term \( a_5 \) is 32 (26+6).

This simple addition process shows how arithmetic sequences grow or shrink steadily with each step, based on the defined common difference.