Problem 69 - Algebra for College Students

Question

Explain the difference between an arithmetic sequence and a geometric sequence.

Step-by-Step Solution

Answer

In arithmetic sequences, terms follow by addition (common difference); in geometric sequences, by multiplication (common ratio).

Step 1: Define Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, denoted usually as \(d\). For example, in the sequence 3, 6, 9, 12, the common difference \(d\) is 3. The formula for the \(n\)-th term in an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term.

Step 2: Define Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as \(r\). For example, in the sequence 2, 6, 18, 54, the common ratio \(r\) is 3. The formula for the \(n\)-th term in a geometric sequence is \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term.

Step 3: Compare Both Sequences

The key difference between an arithmetic and a geometric sequence is how the terms are determined. In an arithmetic sequence, you add the same value (common difference) to each term. In a geometric sequence, you multiply each term by the same value (common ratio). This means arithmetic sequences change linearly while geometric sequences change exponentially.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is defined as a sequence of numbers where each term after the first is generated by adding a constant value. This constant is referred to as the common difference, denoted by the letter \(d\). For example, consider the sequence 5, 8, 11, 14, where each subsequent number increases by 3. Here, the common difference is 3.
The general formula to find the \(n\)-th term \(a_n\) in an arithmetic sequence is given by:

• \(a_n = a_1 + (n-1) \cdot d\)

where \(a_1\) is the first term of the sequence, \(n\) is the term number, and \(d\) is the common difference.
Arithmetic sequences are straightforward as they involve simple addition to advance from one term to the next.

Geometric Sequence

A geometric sequence consists of numbers where each new term after the first is found by multiplying the previous one by a constant known as the common ratio. This common ratio is typically denoted as \(r\). If we consider the sequence 2, 6, 18, 54, we notice that every term is obtained by multiplying the former by 3, so the common ratio is 3.
To find the \(n\)-th term \(a_n\) of a geometric sequence, we use:

• \(a_n = a_1 \cdot r^{(n-1)}\)

where \(a_1\) is the first term, and \(r\) is the common ratio.
Geometric sequences exhibit exponential growth or decay depending on whether the common ratio is greater or less than 1.

Common Difference

The common difference in an arithmetic sequence is the number you add to each term to get to the next. It's the backbone of the arithmetic sequence.
In our example sequence 4, 7, 10, 13, the common difference \(d\) is 3. This signifies a consistent shift, creating a linearly expanding or shrinking pattern.
When you visualize this sequence, think of evenly spaced steps moving either up or closer to zero. The common difference helps predict future terms in the sequence, provided the pattern remains continuing.

Common Ratio

The common ratio is central to understanding geometric sequences. It's the constant factor you multiply each term by to reach the subsequent term. In the sequence 5, 10, 20, 40, the common ratio \(r\) is 2.
This ratio reveals how rapidly the sequence grows or contracts, akin to a rapidly unfolding spiral or a quickly shrinking set of values. The common ratio dictates the nature of change: if \(r > 1\), the sequence grows; if \(0 < r < 1\), it shrinks. Thus, the common ratio encodes the sequence's overall behavior and destiny.