Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Write the first five terms of each arithmetic sequence. Do not use a calculator. The first term is \(-2,\) and the common difference is 12.

Step-by-Step Solution

Answer

The first five terms are -2, 10, 22, 34, and 46.

Step 1: Understanding an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d). In this problem, the common difference is 12 and the first term (a_1) is -2.

Step 2: Writing the Formula for the nth Term

The nth term of an arithmetic sequence can be found using the formula: \( a_n = a_1 + (n-1) \cdot d \). Here, \( a_1 = -2 \) and \( d = 12 \). We'll use this formula to find the first five terms.

Step 3: Calculating the First Term

The first term is given: \( a_1 = -2 \). This term is already known.

Step 4: Calculating the Second Term

Using the formula, calculate the second term: \( a_2 = a_1 + (2-1) \cdot d = -2 + 1 \cdot 12 = 10 \).

Step 5: Calculating the Third Term

Calculate the third term using the formula: \( a_3 = a_1 + (3-1) \cdot d = -2 + 2 \cdot 12 = 22 \).

Step 6: Calculating the Fourth Term

Calculate the fourth term with the formula: \( a_4 = a_1 + (4-1) \cdot d = -2 + 3 \cdot 12 = 34 \).

Step 7: Calculating the Fifth Term

Calculate the fifth term using the formula: \( a_5 = a_1 + (5-1) \cdot d = -2 + 4 \cdot 12 = 46 \).

Step 8: Listing the First Five Terms

The first five terms of the sequence are: \(-2, 10, 22, 34, 46\).

Key Concepts

Common Difference

In the world of arithmetic sequences, the common difference is a key player. It tells you how much to add (or subtract) each time you move to the next term in the sequence.
To put it simply, it is the fixed number that is added to each term to get the next term. For example, if your common difference is 12, like in our exercise, you add 12 to the first term to get the second term, and then again add 12 to get the third term, and so on.

• Example: If the first term is -2 and the common difference is 12, the sequence proceeds as -2, 10, 22, 34, and so forth.

Without the common difference, each term could randomly differ from the last one, and it wouldn't form an arithmetic sequence. Remember, in arithmetic sequences, the difference stays constant, making it predictable and easy to extend.

Nth Term Formula

Finding a specific term in an arithmetic sequence is a breeze with the nth term formula. This formula gives you a direct way to know any term of the sequence without listing all preceding terms.
The nth term of an arithmetic sequence is given by: \[ a_n = a_1 + (n-1) \cdot d \]where

• \(a_n\) is the term you want to find
• \(a_1\) is the first term of the sequence
• \(d\) is the common difference
• \(n\) is the number of the term in the sequence

Let's say the first term is -2 and the common difference is 12. Then, to find the second term, plug \(n=2\) into the formula and calculate:\[ a_2 = -2 + (2-1) \cdot 12 = 10 \]Each new term can be found quickly, and you can see how the formula simplifies the process.

Sequences and Series

Before diving deep into the calculation, it's important to understand what sequences and series are. A sequence is simply a list of numbers arranged in a specific order, following a particular pattern.
In arithmetic sequences, each number after the first is obtained by adding the common difference to the previous number. - For example, the series \(-2, 10, 22, 34, 46\) is created by repeatedly adding 12 after the first term.A series, on the other hand, often refers to the sum of the terms of a sequence. While our focus here is on sequences, series start to play a critical role when you're adding all terms of a sequence up to a certain point, which will be covered more as you advance in topics.