Problem 22 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Can these terms be the terms of an arithmetic sequence? If so, find the common difference. $$2,4,6,8, \dots$$

Step-by-Step Solution

Answer

Yes, it is an arithmetic sequence with a common difference of 2.

Step 1: Understanding Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant, called the common difference. For a sequence \( a_1, a_2, a_3, ... \), the common difference is given by \( d = a_{n+1} - a_n \).

Step 2: Calculate the Difference Between First Pair of Terms

Subtract the first term, \( a_1 = 2 \), from the second term, \( a_2 = 4 \). \[d_1 = a_2 - a_1 = 4 - 2 = 2\]

Step 3: Calculate the Difference Between Second Pair of Terms

Subtract the second term, \( a_2 = 4 \), from the third term, \( a_3 = 6 \). \[d_2 = a_3 - a_2 = 6 - 4 = 2\]

Step 4: Calculate the Difference Between Third Pair of Terms

Subtract the third term, \( a_3 = 6 \), from the fourth term, \( a_4 = 8 \). \[d_3 = a_4 - a_3 = 8 - 6 = 2\]

Step 5: Verify Consistency of Common Difference

Compare all calculated differences: \( d_1 = 2 \), \( d_2 = 2 \), \( d_3 = 2 \). Since they are all the same, the sequence is arithmetic with a common difference of 2.

Key Concepts

Common Difference

In an arithmetic sequence, the magic that holds everything together is the constant gap between consecutive numbers, known as the common difference. This is the number you are adding (or subtracting) every time you move from one term to the next.
To find this difference, you simply subtract any term from the term that comes right after it.

• If you have a sequence like 2, 4, 6, 8, you determine the common difference by calculating \( 4 - 2 \), \( 6 - 4 \), and \( 8 - 6 \). You find that each equals 2.
• It’s important that these differences stay the same throughout to qualify as an arithmetic sequence.

The arithmetic sequence with a common difference is predictable, making it easy to figure out what numbers will come next.

Sequence

A sequence is basically a list of numbers in a specific order. Each number in the list is called a term. In the context of arithmetic sequences, the order is crucial because each term needs to follow the rule of the common difference.
Not all sequences are arithmetic. An arithmetic sequence is special because every term after the first is formed by adding the common difference to the previous term.

• For example, in our sequence 2, 4, 6, 8, each new number is found by adding the common difference of 2 to the last number.
• This means arithmetic sequences are linear and predictable.

Recognizing this pattern can help you quickly identify and work with arithmetic sequences.

Arithmetic Series

When you add up the terms of an arithmetic sequence, you get an arithmetic series. The series is the sum of all the numbers in that sequence. Imagine you want to find out the total of numbers like 2, 4, 6, and 8. Instead of adding each one individually, there are formulas to make this task easier.
The sum of the first \( n \) terms of an arithmetic sequence can be found using the formula:
\[S_n = \frac{n}{2} (a_1 + a_n)\]
- \( S_n \) represents the sum of the first \( n \) terms.- \( a_1 \) is the first term, and \( a_n \) is the nth term in the sequence.Using these elegant tools at your disposal, calculating the total value of an arithmetic series becomes much simpler. Understanding this can be quite useful in various mathematical problems and real-life scenarios.