Problem 8 - 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 \(-2,\) and the common difference is 12

Step-by-Step Solution

Answer

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

Step 1: Identify the Arithmetic Sequence Formula

The formula for the nth term of an arithmetic sequence is given by \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term, and \( d \) is the common difference.

Step 2: Calculate the Second Term

Using the formula, substitute \( n = 2 \), \( a_1 = -2 \), and \( d = 12 \). Thus, \( a_2 = -2 + (2-1) imes 12 = 10 \).

Step 3: Calculate the Third Term

Using the formula with \( n = 3 \), calculate \( a_3 = -2 + (3-1) imes 12 = 22 \).

Step 4: Calculate the Fourth Term

Using the formula with \( n = 4 \), calculate \( a_4 = -2 + (4-1) imes 12 = 34 \).

Step 5: Calculate the Fifth Term

Using the formula with \( n = 5 \), calculate \( a_5 = -2 + (5-1) imes 12 = 46 \).

Key Concepts

Common Difference

In the world of arithmetic sequences, the term "common difference" is very important. It is the difference between consecutive terms in a sequence. This value stays the same for every pair of successive terms. Understanding it is essential for predicting the sequence's growth or decrease.
For example, in the arithmetic sequence starting with -2, and a common difference of 12, you will always add 12 to get from one term to the next. This arithmetic pattern allows you to easily extend the sequence from any point.
Knowing the common difference helps you easily build sequences by repeatedly adding (or subtracting) this number. If the common difference is positive, the sequence increases. If it's negative, the sequence decreases. Simple, right? That's the magic of arithmetic sequences!

Nth Term Formula

When working with arithmetic sequences, the Nth term formula is your best friend for quickly finding a specific term number in the list. The formula looks like this: \[ a_n = a_1 + (n-1) \times d \] where:

• \(a_n\) is the Nth term
• \(a_1\) is the first term
• \(n\) is the term number
• \(d\) is the common difference

This formula neatly identifies any term in an arithmetic sequence without needing to calculate all previous terms. With this formula, you can jump directly to any term number you want.
Imagine you want to find the fifth term of our sequence starting with -2 and a common difference of 12. By substituting into the formula, you get: \[ a_5 = -2 + (5-1) \times 12 = 46 \] Now you know the fifth term without arduous calculations!

Sequence Calculation

Once you've got a handle on the common difference and the Nth term formula, calculating the terms of an arithmetic sequence becomes a breeze. Start with the first term, and use the common difference to progress through the sequence.
For example, starting with our sequence of -2 and a common difference of 12, use the Nth term formula to calculate each term one by one:

• The first term is simply the initial term, \(-2\).
• The second term: \(-2 + 12 = 10\).
• The third term: \(-2 + 24 = 22\).
• The fourth term: \(-2 + 36 = 34\).
• The fifth term: \(-2 + 48 = 46\).

As you can see, you repeatedly add the common difference to know each next number. This repetition of adding 12 takes us smoothly through the pattern of this sequence.