Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=3(n-1)+5$$

Step-by-Step Solution

Answer

The first five terms are 5, 8, 11, 14, and 17.

Step 1: Understand the Formula

We are given the sequence formula \(a_{n}=3(n-1)+5\). This formula determines each term of the sequence using the variable \(n\), which represents the position number of the term in the sequence.

Step 2: Calculate the First Term

To find the first term \(a_1\), we substitute \(n=1\) into the formula: \[a_1 = 3(1-1) + 5 = 3(0) + 5 = 5\]. So, the first term is 5.

Step 3: Calculate the Second Term

Substitute \(n=2\) to find the second term: \[a_2 = 3(2-1) + 5 = 3(1) + 5 = 3 + 5 = 8\], giving us a second term of 8.

Step 4: Calculate the Third Term

Substitute \(n=3\) to get the third term: \[a_3 = 3(3-1) + 5 = 3(2) + 5 = 6 + 5 = 11\]. Thus, the third term is 11.

Step 5: Calculate the Fourth Term

Substitute \(n=4\) to determine the fourth term: \[a_4 = 3(4-1) + 5 = 3(3) + 5 = 9 + 5 = 14\]. The fourth term is 14.

Step 6: Calculate the Fifth Term

Substitute \(n=5\) into the formula for the fifth term: \[a_5 = 3(5-1) + 5 = 3(4) + 5 = 12 + 5 = 17\]. Therefore, the fifth term is 17.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers with a constant difference between consecutive terms. This difference is referred to as the "common difference". If you're given an arithmetic sequence, you'll notice each number forms a predictable pattern. For example, if the common difference is 3, each term increases by 3 when moving from one term to the next. This linear growth makes arithmetic sequences easy to recognize and understand.

• An example sequence might be: 5, 8, 11, 14, 17.
• Here, the difference between each term is 3, demonstrating the constant common difference.

The simplicity of arithmetic sequences makes them useful for solving real-world problems quickly and efficiently. You can spot them in many everyday scenarios, such as scheduling tasks or distributing items evenly.

Formula

In arithmetic sequences, the formula plays a crucial role in finding any term in the sequence without listing all prior terms. The general form of an arithmetic sequence formula is: \[ a_n = a_1 + (n-1) imes d \] where:

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

This formula is powerful because it allows you to calculate any term in the sequence directly, saving time and effort when dealing with long sequences. When used correctly, it underscores the predictable nature of arithmetic sequences.

Term Calculation

Calculating terms in an arithmetic sequence follows a straightforward process. Let's take the given formula from the exercise: \[ a_{n} = 3(n-1) + 5 \] The process involves substituting different values of \(n\) (position number) to determine various terms:

• For the first term, substitute \(n = 1\): \(a_1 = 3(1-1) + 5 = 5\).
• For the second term, use \(n = 2\):\(a_2 = 3(2-1) + 5 = 8\).
• Continuing this way, if \(n = 3\), then \(a_3 = 11\).
• The pattern continues such that each subsequent substitution increases the term by the common difference.

This repeated substitution based on the value of \(n\) illustrates the predictable step-by-step calculation involved in generating terms in the sequence.

Position Number

The position number, often represented by \(n\), plays a vital role in understanding arithmetic sequences. It tells you where a specific term falls in the sequence. Using the formula \(a_{n} = 3(n-1) + 5\), the position number directly influences the term calculation.

• For example, with \(n = 1\), we determine the first term.
• With \(n = 2\), we calculate the second term, and so forth.
• Each step from one position number to the next involves adding the common difference to the previous term.

The concept of position numbers is central because it helps you understand how far into the sequence a particular term is. Once you grasp this, you can easily navigate through the sequence and calculate terms with confidence. Knowing the position number is especially handy when dealing with very large or very small terms in an arithmetic sequence.