Problem 28 - Algebra 2

Question

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 45,90,180,360, \dots $$

Step-by-Step Solution

Answer

The sequence is geometric, with a common ratio of 2. The next two terms of the sequence are 720 and 1440.

Step 1: Observing the Sequence

The first step is observing how the numbers in the sequence progress. Here there are clearly not constant differences between terms, so the sequence is not arithmetic. However, each term appears to be twice the previous term, suggesting it might be a geometric sequence.

Step 2: Checking for a Common Ratio

To confirm whether it's a geometric sequence, divide each term by the previous term. Doing this for every pair of consecutive terms we get: \(90/45 = 2, 180/90 = 2, 360/180 = 2\), which confirms that all have a common ratio, and are therefore part of a geometric sequence.

Step 3: Finding the Next Two Terms

Since this is confirmed as a geometric sequence, and we've identified the common ratio as 2, we simply multiply the last given term by the common ratio to find the next terms. Hence, \(360 \times 2 = 720\), and \(720 \times 2 = 1440\). So, 720 and 1440 are the next two terms.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a series of numbers where each term after the first is created by adding a constant, known as the "common difference," to the previous term. For example, in the sequence 2, 4, 6, 8, ... each number increases by 2, which is the common difference.

• Constant Addition: The common difference means you add the same number to each term to get to the next one. If the difference isn't the same between the terms, it's not an arithmetic sequence.
• Formula: The general formula for an arithmetic sequence is given by \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term, \(n\) is the term number, \(d\) is the common difference, and \(a_n\) is the \(n\)-th term.

The terms in the sequence above (45, 90, 180, 360, ...) do not have a consistent common difference. Therefore, this sequence is not arithmetic.

Common Ratio

In contrast to arithmetic sequences, a geometric sequence involves multiplying each term by a constant to get the next term. This constant is known as the "common ratio." In the sequence given as 45, 90, 180, 360, ..., each term is multiplied by 2 to get the next, making 2 the common ratio.

• Determining the Common Ratio: To find the common ratio, divide a term by the previous term. For a sequence to be geometric, this ratio should be the same throughout.
• Example: In the provided sequence, we can check: \( \frac{90}{45} = 2\), \( \frac{180}{90} = 2\), and \( \frac{360}{180} = 2\). Therefore, the common ratio is indeed 2.

Identifying the common ratio is key to understanding geometric progressions and predicting subsequent terms.

Sequence Progression

Sequence progression is all about understanding how each term in a sequence relates to the others. Identifying whether a sequence is arithmetic, geometric, or neither helps determine how the sequence will progress.

• Arithmetic Progression: Each term is the previous term plus the common difference.
• Geometric Progression: Each term is the previous term multiplied by the common ratio.

To find the next terms in a sequence—such as in a geometric sequence—first confirm the type of sequence and the value of the common ratio or difference. For the sequence 45, 90, 180, 360, ..., since it is geometric with a common ratio of 2, the next term is calculated by multiplying 360 by 2, giving us 720, and then multiplying 720 by 2 to get 1440.

Predicting sequence progression is useful across different areas, from mathematics to understanding patterns in science and technology.