Problem 61 - Algebra for College Students

Question

Which of the following sequences is not an arithmetic sequence? Explain your answer. a) \(\frac{1}{2}, 1, \frac{3}{2}, \dots\) b) \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\) c) \(5,0,-5, \dots\) d) \(2,3,4, \ldots\)

Step-by-Step Solution

Answer

Sequence b is not an arithmetic sequence.

Step 1 - Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same, called the common difference.

Step 2 - Check Sequence a

Calculate the differences between consecutive terms: \ \ \(1 - \frac{1}{2} = \frac{1}{2}\text{ and }\frac{3}{2} - 1 = \frac{1}{2}\text{. The common difference is }\frac{1}{2}\text{. So, this is an arithmetic sequence.}\)

Step 3 - Check Sequence b

Calculate the differences between consecutive terms: \ \ \( \frac{1}{3} - \frac{1}{2} = \frac{-1}{6}\text{ and }\frac{1}{4} - \frac{1}{3} = \frac{-1}{12}\text{. Since the common difference is not constant, this is not an arithmetic sequence.}\)

Step 4 - Check Sequence c

Calculate the differences between consecutive terms: \ \ \( 0 - 5 = -5 \text{ and } (-5) - 0 = -5\text{. The common difference is }-5\text{. So, this is an arithmetic sequence.}\)

Step 5 - Check Sequence d

Calculate the differences between consecutive terms: \ \ \( 3 - 2 = 1 \text{ and }4 - 3 = 1\text{. The common difference is }1\text{. So, this is an arithmetic sequence.}\)

Key Concepts

common difference

The common difference is a key concept in understanding arithmetic sequences. It is the difference between any two consecutive terms in the sequence. For a sequence to be classified as arithmetic, this difference must remain constant throughout. To find it, simply subtract an earlier term from the term that follows it. For example, in the sequence 2, 5, 8, 11,..., the common difference is found by calculating:
5 - 2 = 3
8 - 5 = 3
11 - 8 = 3
Since the difference is constant, the common difference in this example is 3. It's essential to always ensure that this difference doesn't change if you're identifying an arithmetic sequence.

consecutive terms

In any sequence, consecutive terms are terms that come one after another without any terms in between. For instance, in the sequence 5, 0, -5,..., the terms 5 and 0, as well as 0 and -5, are considered consecutive.
To determine if a sequence is arithmetic, you look at the consecutive terms and calculate their common difference. This type of check helps in verifying the arithmetic nature of the sequence.
If the difference between consecutive terms remains constant like it does in the following sequences:
2, 4, 6,... (common difference of 2), and
-1, -4, -7,... (common difference of -3), then the sequences are arithmetic.

algebraic sequence

An algebraic sequence is a sequence of terms defined by an algebraic formula. In the case of arithmetic sequences, the formula involves linear expressions. The general formula for an arithmetic sequence is given by:
a_n = a_1 + (n - 1)d Where:

• a_n is the nth term of the sequence
• a_1 is the first term
• n is the position of the term in the sequence
• d is the common difference

For example, to find the 5th term in the sequence 2, 5, 8,... with common difference 3, you'd calculate:
a_5 = 2 + (5 - 1) * 3 = 2 + 12 = 14
This formula ensures that every term in the sequence aligns with the pattern dictated by the common difference.