Problem 60 - College Algebra with Corequisite Support

Question

Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.

Step-by-Step Solution

Answer

Yes, a constant sequence like 1, 1, 1,... is both arithmetic and geometric.

Step 1: Understand the Definitions

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. A geometric sequence is one where the ratio between consecutive terms is constant.

Step 2: Set Conditions for Both

To satisfy both conditions simultaneously, the difference between terms (for arithmetic) must be constant, and the ratio (for geometric) must be constant.

Step 3: Investigate Simple Sequences

Consider simple sequences, for example: a constant sequence. The sequence 1, 1, 1, 1,... where each term is 1, has both a common difference of 0 (arithmetic) and a common ratio of 1 (geometric).

Step 4: Verify Conditions

Check if the sequence identified fulfills both conditions. For 1, 1, 1, the difference between any two consecutive terms is 0 (arithmetic condition), and the ratio is 1 (geometric condition). This verifies that the sequence is both arithmetic and geometric.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is derived by adding a fixed, constant value to the previous term. This value is called the "common difference". For example, if you have a sequence starting with 2 and a common difference of 3, the sequence would be 2, 5, 8, 11, and so on. In more formal terms, the nth term of an arithmetic sequence can be calculated using the formula:

• \( a_n = a_1 + (n-1) \, d \)

where \( a_1 \) is the first term and \( d \) is the common difference. The charm of arithmetic sequences is their straightforward incremental pattern. Each step forward is a simple leap by the exact same amount, making predictions and finding terms relatively easy. Understanding arithmetic sequences can help solve problems related to regularly spaced increases or decreases.

Geometric Sequence

A geometric sequence involves numbers that are multiplied by a constant value to get the next term. This constant is known as the "common ratio". Let's consider a sequence beginning with the number 3 and a common ratio of 2. This sequence would be 3, 6, 12, 24, and it would continue multiplying by the common ratio each step. To find the nth term of a geometric sequence, the formula is:

• \( a_n = a_1 \times r^{(n-1)} \)

where \( a_1 \) is the first term and \( r \) is the common ratio. Geometric sequences are particularly fascinating because they can model exponential growth or decay. They are useful in various fields, including finance for interest calculations and science for population modeling.

Common Difference

The common difference is a key feature of an arithmetic sequence and is responsible for the pattern of the sequence. For an arithmetic sequence, subtract the previous term from the current term to find the common difference. For example, take the sequence 4, 7, 10, 13, where each time you add 3 to get the next term. This 3 is the common difference. Knowing the common difference immediately gives you insight into how the sequence generates subsequent terms. In equations, it simplifies calculations by offering a constant integer or decimal to work with, streamlining predictions and helping identify specific terms without listing them all.

Common Ratio

The common ratio is what defines the multiplicative pattern in a geometric sequence. To find it, divide any term by the previous term. For instance, if your sequence is 5, 10, 20, 40, the common ratio is found by dividing 10 by 5, resulting in a ratio of 2. This ratio is consistently used to multiply to get from one term to the next.

• The formula for a geometric sequence heavily relies on the common ratio: \( a_n = a_1 \times r^{(n-1)} \).

The common ratio is pivotal in modeling and understanding various real-world phenomena, such as the doubling pattern of bacteria growth or the decay of radioactive materials. Appreciating how the common ratio operates allows us to anticipate the sequence's behavior even for large numbered terms without computing every prior term.