Problem 33 - Precalculus with Limits

Question

In Exercises 33 - 40, write the first five terms of the arithmetic sequence. \( a_1 = 5, d = 6 \)

Step-by-Step Solution

Answer

The first five terms of the arithmetic sequence are 5, 11, 17, 23, 29

Step 1: Finding the First Term

The first term \(a_1\) of the arithmetic sequence is given as 5

Step 2: Finding the Second Term

The second term of an arithmetic sequence is found by adding the common difference \(d\) to the first term \(a_1\). Therefore, the second term \(a_2\) is \(a_1 + d = 5 + 6 = 11\)

Step 3: Finding the Third Term

The third term of the arithmetic sequence is found by adding the common difference \(d\) to the second term \(a_2\). Therefore, the third term \(a_3\) is \(a_2 + d = 11 + 6 = 17\)

Step 4: Finding the Fourth Term

The fourth term of an arithmetic sequence is found by adding the common difference \(d\) to the third term \(a_3\). Therefore, the fourth term \(a_4\) is \(a_3 + d = 17 + 6 = 23\)

Step 5: Finding the Fifth Term

The fifth term of an arithmetic sequence is found by adding the common difference \(d\) to the fourth term \(a_4\). Therefore, the fifth term \(a_5\) is \(a_4 + d = 23 + 6 = 29\)

Key Concepts

Arithmetic Progression

An arithmetic progression, also called an arithmetic sequence, is a sequence of numbers in which the difference between any two consecutive terms is always the same. This constant difference is what characterizes an arithmetic progression. Such sequences are common in algebra and are easy to work with once you understand the basic principle.

The main properties of an arithmetic progression include:

• A linear and consistent increase or decrease between terms.
• The ability to express any term of the sequence using a formula.
• Each term can be obtained by adding the common difference to the previous term, starting from the given initial term.

The general formula for the nth term in an arithmetic progression is:\[ a_n = a_1 + (n - 1) imes d \]where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) represents the term number. Understanding these properties and this formula allows you to easily discover any term in the sequence, regardless of how far along it may be.

Common Difference

The common difference in an arithmetic sequence is a crucial component. It is the fixed amount that you add (or subtract if negative) to find the next term in the sequence. Understanding how this difference works is key to mastering arithmetic sequences.

The common difference \(d\) is calculated by subtracting the first term from the second term, giving:\[ d = a_{n+1} - a_n \]where \(a_n\) is any term and \(a_{n+1}\) is the subsequent term.

In the original problem, the common difference is given as 6. This means that each term increases by 6 to reach the next term. Knowing the common difference allows you to quickly find subsequent terms in the sequence.

Sequence Terms

Sequence terms are simply the individual elements that make up a sequence. In arithmetic sequences, each term can be calculated using an initial term and the common difference.

Let's look at the problem:

• The first term \(a_1\) is 5.
• The second term, \(a_2\), is calculated as \(a_1 + d = 5 + 6 = 11\).
• The third term, \(a_3\), uses the second term: \(a_2 + d = 11 + 6 = 17\).
• The fourth term, \(a_4\), follows likewise: \(a_3 + d = 17 + 6 = 23\).
• Finally, the fifth term \(a_5\) is: \(a_4 + d = 23 + 6 = 29\).

In this sequence, each term builds directly upon the one before it, consistently moving forward by the common difference. By understanding how sequence terms work, you can efficiently determine each term in the sequence given the common difference and the initial term.