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Chapter 4: Problem 18

Find the next three terms of each sequence. \(1,2,4,8, \ldots\)

Short Answer

Expert verified
The next three terms are 16, 32, and 64.

Step by step solution

01

Identify the Pattern

Observe the sequence given: 1, 2, 4, 8. Notice that each term after the first is obtained by multiplying the previous term by 2. Therefore, the pattern of the sequence is that it is a geometric sequence with a common ratio of 2.
02

Apply the Pattern to Find the Next Term

To find the next term in the sequence, multiply the last known term (8) by the common ratio (2). The calculation is as follows:\( 8 \times 2 = 16 \).Thus, the next term is 16.
03

Find the Second Next Term

Using the pattern, multiply the most recent term (16) by 2 to find the second next term:\( 16 \times 2 = 32 \).The second next term is 32.
04

Find the Third Next Term

Finally, multiply 32 (the most recent term) by the common ratio 2 to get the third next term:\( 32 \times 2 = 64 \).The third next term is 64.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pattern Recognition
Recognizing patterns is a crucial skill in understanding sequences. When you look at a list of numbers such as 1, 2, 4, 8, you need to identify what changes from one term to the next.
In this example, noticing that each term is double the previous one helps you see that there is a multiplying pattern involved.
By observing this pattern, we can determine that this sequence is not just random numbers, but follows a systematic approach where each term is obtained by multiplying the one before it.
Common Ratio
The common ratio is the factor that links consecutive terms in a geometric sequence. In our sequence, 1, 2, 4, 8, the common ratio is 2.
This means that every term is the previous term multiplied by 2. The common ratio is consistent throughout the sequence, making it predictable and allowing you to easily generate future terms.
Whenever you encounter a potential geometric sequence, a good first step is to test if there's a common ratio by dividing one term by the previous term to see if you get the same result consistently (in this case, 2).
Sequence Terms
Sequence terms are simply the individual numbers that make up a sequence. For example, in our sequence 1, 2, 4, 8, 16, 32, and so on, each of these numbers is a term.
Understanding how each term relates to its predecessors and successors is key to mastering sequences.
Once you have identified the pattern and the common ratio, finding new terms becomes straightforward as you keep applying the same rule鈥攎ultiply by the common ratio to transition from one term to the next in a geometric sequence.

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