Problem 72
Question
Determine whether the sequence is arithmetic, geometric, or neither. $$66,55,44,33,22, \dots$$
Step-by-Step Solution
Verified Answer
The sequence \(66, 55, 44, 33, 22, \dots\) is an arithmetic sequence.
1Step 1: Determine arithmetic sequence
First, find if the sequence is arithmetic by subtracting subsequent terms. This is done as follows: \(55 - 66 = -11\), \(44 - 55 = -11\), \(33 - 44 = -11\), and \(22 - 33 = -11\) . Since each difference is the same (-11), this indicates that the sequence could be arithmetic.
2Step 2: Determine geometric sequence
Next, find if the sequence is geometric by dividing subsequent terms. This is done as follows: \(\frac{55}{66} \approx 0.83\), \(\frac{44}{55} \approx 0.80\), \(\frac{33}{44} \approx 0.75\), and \(\frac{22}{33} \approx 0.67\). Since the ratios are not equal, this suggests that the sequence is not geometric.
3Step 3: Interpret results
Once both an arithmetic and geometric sequence have been examined, the type of sequence can be determined. Since we obtained constant differences and variable ratios, the sequence is characterized as arithmetic and not geometric.
Key Concepts
Sequence in MathematicsArithmetic ProgressionGeometric Sequence
Sequence in Mathematics
A sequence in mathematics is a collection of numbers arranged in a specific order that follows a particular rule. Sequences are fundamental in mathematics because they can represent patterns or forms of counting, and are often used to introduce the concept of a function. They're present in various aspects of mathematics, from simple counting to more complex concepts like series and convergence.
There are many types of sequences, but two very common ones are arithmetic and geometric sequences. These sequences are simple yet powerful tools for solving problems in algebra, prediction, and other areas of applied mathematics. Understanding the nature of a sequence is crucial for students to make predictions about future terms or identify the structure of the sequence for further mathematical analysis.
There are many types of sequences, but two very common ones are arithmetic and geometric sequences. These sequences are simple yet powerful tools for solving problems in algebra, prediction, and other areas of applied mathematics. Understanding the nature of a sequence is crucial for students to make predictions about future terms or identify the structure of the sequence for further mathematical analysis.
Arithmetic Progression
An arithmetic progression, also known as an arithmetic sequence, is a series of numbers in which the difference between consecutive terms is constant. This constant is called the common difference. A classic example of an arithmetic progression is a sequence of odd numbers, for instance, 1, 3, 5, 7, where the common difference is 2.
To identify an arithmetic progression, one can simply subtract any term from the subsequent term. If the result is the same for every pair of consecutive terms, then the sequence is indeed arithmetic. For example, in the sequence provided in the exercise, the subtraction of any term from the following term yields a consistent result of -11, clearly indicating that the sequence is an arithmetic progression.
To identify an arithmetic progression, one can simply subtract any term from the subsequent term. If the result is the same for every pair of consecutive terms, then the sequence is indeed arithmetic. For example, in the sequence provided in the exercise, the subtraction of any term from the following term yields a consistent result of -11, clearly indicating that the sequence is an arithmetic progression.
Properties of Arithmetic Progression
- The difference between consecutive terms is constant and equal to the common difference.
- The nth term of an arithmetic sequence can be found using the formula: \( a_n = a_1 + (n-1)d \) where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( d \) is the common difference.
- Sum of the first n terms (\( S_n \) ) of an arithmetic sequence can be calculated using the formula: \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \)
Geometric Sequence
A geometric sequence, on the other hand, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Examples of geometric sequences include the powers of 2: 2, 4, 8, 16, or the sequence 3, 6, 12, 24, where the common ratio is 2.
To check if a sequence is geometric, divide each term by the previous term. If the result, the common ratio, is the same for all consecutive pairs of terms, then the sequence is geometric. Taking the sequence from the exercise, and dividing each term by its predecessor, we find that the ratios are not the same, thus the sequence is not geometric.
To check if a sequence is geometric, divide each term by the previous term. If the result, the common ratio, is the same for all consecutive pairs of terms, then the sequence is geometric. Taking the sequence from the exercise, and dividing each term by its predecessor, we find that the ratios are not the same, thus the sequence is not geometric.
Characteristics of Geometric Sequence:
- The ratio between consecutive terms is constant and is known as the common ratio.
- The nth term of a geometric sequence is found with the formula \( a_n = a_1 \times r^{(n-1)} \) where \( a_1 \) is the first term and \( r \) is the common ratio.
- The sum of the first n terms (\( S_n \) ) of a geometric sequence can be calculated using the formula: \( S_n = a_1 \frac{1 - r^n}{1 - r} \) for \( r \) not equal to 1.
Other exercises in this chapter
Problem 72
Find the zeros (if any) of the rational function. $$f(x)=6+\frac{4}{x^{2}+4}$$
View solution Problem 72
Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y^{2}-4 y-4 x=0$$
View solution Problem 73
Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$y=e^{x}$$
View solution Problem 73
Find the zeros (if any) of the rational function. $$f(x)=5-\frac{3}{x-2}$$
View solution