Problem 35 - Algebra 2

Question

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 2,2,2,2, \dots $$

Step-by-Step Solution

Answer

The sequence (2,2,2,2) can be identified as both an arithmetic and geometric sequence. The next two terms of the sequence are 2 and 2.

Step 1: Determine Type of Sequence

Since every term in the sequence is the same (2), the difference between every two consecutive terms is 0. An arithmetic sequence is defined by common difference. Here, the difference is consistent (0), so this is an arithmetic sequence. However, given the terms are also the same, it can also be identified as a geometric sequence where the ratio of any two successive terms is 1 (since 2/2 = 1). The ratio is constant, and hence it fits the definition of a geometric sequence too. Therefore, the sequence can be both arithmetic or geometric.

Step 2: Define Terms and Sequence

An arithmetic sequence can be described as \( a, a + d, a + 2d, a + 3d, .. \) where \( a \) is the first term and \( d \) is the common difference. Here in the sequence (2, 2, 2, 2), \( a = 2 \) and \( d = 0 \). A geometric sequence can be defined as \( a, ar, ar^2, ar^3, .. \) where \( a \) is the first term, \( r \) is the common ratio. Here, since \( a = 2 \) and \( r = 1 \), this sequence can also be presented as (2, 2x1, 2x1^2). Thus, it can be identified as both arithmetic and geometric.

Step 3: Find the Next Two Terms

Since the arithmetic sequence's difference or the geometric sequence's ratio is constant, the next two terms will be the same as the previous terms. Therefore, the next two terms are also 2 and 2.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a type of number sequence where each term is generated by adding a constant value to the previous term. This constant value is known as the common difference.
For example, the sequence 3, 6, 9, 12, ... is arithmetic because each term increases by a common difference of 3. Let's break down how we identify an arithmetic sequence:

• The sequence must add a consistent amount to each term.
• It takes the form: \( a, a + d, a + 2d, a + 3d, \ldots \), where \( a \) is the first term and \( d \) is the common difference.

In the case of the sequence 2, 2, 2, 2, ..., since adding zero to every term results in the same term, the common difference \( d \) is 0. This is why the sequence is also considered arithmetic.

Geometric Sequence

A geometric sequence is a series of numbers where each term is derived by multiplying the previous term by a fixed, non-zero number known as the common ratio.
Consider the sequence 2, 4, 8, 16, ..., which is geometric because each term is obtained by multiplying the previous term by a common ratio of 2.

• The multiplication factor remains constant between terms.
• The sequence follows the pattern: \( a, ar, ar^2, ar^3, \ldots \), where \( a \) is the first term and \( r \) is the common ratio.

In our example, 2, 2, 2, 2, ..., the common ratio is 1 since multiplying each term by 1 keeps it unchanged.

Common Difference

The common difference is a key feature of an arithmetic sequence. It indicates the constant amount added to each term to produce the next.
This difference can be calculated by subtracting any term in the sequence from the one that follows it.

• If the difference remains the same for all consecutive terms, the sequence is confirmed to be arithmetic.
• For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3 (since 5 - 2 = 3).

In our 2, 2, 2, 2, ..., example, the common difference is 0, meaning each term is unchanged from the previous term, characterizing it as an arithmetic sequence.

Common Ratio

The common ratio is a pivotal concept in geometric sequences. It represents the fixed factor by which each term is multiplied to get to the next term.
You find the common ratio by dividing any term by the previous term.

• If this ratio is constant across all pairs of terms, the sequence qualifies as geometric.
• For instance, in the sequence 3, 9, 27, ..., dividing each term by its predecessor yields a common ratio of 3.

In the sequence 2, 2, 2, 2, ..., any term divided by its previous term yields a common ratio of 1, making it geometric because the ratio remains constant throughout.