Problem 1 - Algebra 2

Question

Is the sequence geometric? If so, find the common ratio and the next two terms. $$ 1,2,4,8, \ldots $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of 2. The next two terms are 16 and 32.

Step 1: Identify the sequence

The sequence provided is \(1,2,4,8,\ldots\).

Step 2: Check if it is a geometric sequence

Take any two consecutive terms in the sequence. If the ratio of every two consecutive terms is constant, then the sequence is geometric. For instance, \(2/1 = 2, 4/2 = 2, 8/4 = 2\). The ratio is always 2, so the sequence is geometric.

Step 3: Determine the common ratio

In a geometric sequence, the common ratio is found by dividing any term by the previous term. The common ratio here is 2.

Step 4: Find the next two terms

You can find the next terms in the sequence by multiplying the last term given by the common ratio. The fifth term in the sequence would be \(8*2=16\) and the sixth term would be \(16*2=32\).

Key Concepts

Common Ratio

A geometric sequence is characterized by having a constant ratio between consecutive terms. This consistent factor is called the "common ratio". It tells us how each term relates to the previous one.
In the sequence provided: 1, 2, 4, 8, and so on, we calculate this common ratio by dividing each term by the one that precedes it.

• For 2 and 1: the common ratio is \( \frac{2}{1} = 2 \).
• For 4 and 2: the common ratio is \( \frac{4}{2} = 2 \).
• For 8 and 4: the common ratio is \( \frac{8}{4} = 2 \).

Each of these results is consistent, confirming a common ratio of 2. This ratio crucially defines the sequence as geometric, indicating predictability and regularity in its progression.

Sequence Pattern

Understanding the pattern in a geometric sequence helps predict its behavior. In such sequences, each term is a result of multiplying the previous term by a constant factor, the common ratio.
In our sequence, 1, 2, 4, 8,..., the pattern emerges clearly: start with 1 and multiply by the common ratio, 2, to get the next number. Then repeat this process.
This pattern can be expressed as:

• Start with the first term (1).
• Multiply by the common ratio (2) to get the next term.
• Continue the process for subsequent terms.

Recognizing this pattern provides a neat structure, making sequences easier to work with and extend.

Next Terms

Once the common ratio and sequence pattern are established, finding further terms becomes straightforward.
Continuing from the last known term, 8, in our sequence, we apply the sequence pattern:

• Multiply 8 by the common ratio, 2, to get 16.
• Multiply 16 by the common ratio, 2, to get 32.

Hence, the next two terms in the sequence are 16 and 32.
This method illustrates the simplicity of predicting future terms in geometric sequences, offering a powerful tool for analyzing growth, decay, and other applications.