Problem 5 - Algebra 2

Question

Is the sequence geometric? If so, find the common ratio and the next two terms. $$ 10,4,1.6,0.64, \dots $$

Step-by-Step Solution

Answer

Yes, the sequence is geometric with a common ratio of 0.4. The next two terms in the sequence are 0.256 and 0.1024.

Step 1: Identify the Common Ratio

Divide each term in the sequence by the term that precedes it. For the given sequence, this will be \( \frac{4}{10} \), \( \frac{1.6}{4} \), and \( \frac{0.64}{1.6} \). If these divisions give the same result, it means the sequence is geometric, and this common result will be the common ratio.

Step 2: Calculate the Common Ratio

Performing these divisions, every calculation yields 0.4. Therefore, the sequence is a geometric one, with a common ratio of 0.4.

Step 3: Calculate the next two terms

To find the next terms in the sequence, multiply the latest term in the sequence by the common ratio. So, the next two terms will be \( 0.64 * 0.4 \) and \( (0.64 *0.4) * 0.4 \)

Key Concepts

Common Ratio

In a geometric sequence, the constant factor between consecutive terms is known as the common ratio. It is the ingredient that makes a sequence geometric. To find this ratio, you divide each term in the sequence by the term preceding it. For example, you can quickly check if a series of numbers forms a geometric sequence by calculating this common ratio between terms.

From the given sequence: 10, 4, 1.6, 0.64, ...

• Divide 4 by 10: \( \frac{4}{10} = 0.4 \)
• Divide 1.6 by 4: \( \frac{1.6}{4} = 0.4 \)
• Divide 0.64 by 1.6: \( \frac{0.64}{1.6} = 0.4 \)

Notice that each division results in the same number, 0.4, confirming the sequence is geometric. Therefore, the common ratio for this sequence is 0.4.

Sequence

A sequence in mathematics is an ordered list of numbers. In a geometric sequence, every term is obtained by multiplying the previous one by a fixed number called the common ratio. This type of sequence grows or shrinks exponentially based on the value of the ratio.

To understand this better, let's consider our example: 10, 4, 1.6, 0.64, ... Each term in this sequence is derived by multiplying the prior term by 0.4, which is their common ratio. In turn, this defines the geometric nature of this sequence.

For students encountering sequences, recognizing that the pattern involves multiplying by a constant value can simplify the task of identifying and continuing the pattern.

Mathematical Operations

Mathematical operations like division and multiplication play a crucial role in dealing with sequences, especially geometric ones. To determine the common ratio, you would divide. To extend the sequence, you multiply with the identified common ratio.
Let’s see how these operations work for the given sequence:

• Division was used to find the common ratio: \( 10 \rightarrow 4 \rightarrow 1.6 \rightarrow 0.64 \). This was done by dividing each term by the previous term resulting in 0.4.
• Multiplication was used to find the next terms: Start with the last known term, 0.64, and multiply by 0.4 to get the next term, 0.256. Multiply 0.256 by 0.4 to get 0.1024.

Mastering these operations is key, as they help you to navigate through and extend sequences, ensuring a deeper understanding of their structure and progression.