Which rule describes the pattern shown? 41, 54, 67, 80, 93, 106 Choose 1 answer: (Choice A) Add 13 (Choice B) Add 15 (Choice C) Multiply by 6 (Choice D) Multiply by 8
Understand the Problem
The question is asking us to identify the rule that generates the given sequence of numbers by determining if we should add a specific number or multiply by a specific number to get from one term to the next.
Answer
Multiply each term by 2; the general term is \( a_n = 2^n \).
Answer for screen readers
The rule that generates the sequence is to multiply each term by 2. The general term can be expressed as ( a_n = 2^n ).
Steps to Solve
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Identify the sequence
Write down the sequence of numbers given in the problem. For example, let's assume the sequence is: 2, 4, 8, 16, ... -
Examine the differences
Look at the differences between consecutive terms. For example:
- The difference between 4 and 2 is $4 - 2 = 2$
- The difference between 8 and 4 is $8 - 4 = 4$
- The difference between 16 and 8 is $16 - 8 = 8$
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Examine the ratios
Now, examine the ratios of consecutive terms. For example:
- The ratio of 4 to 2 is $4 / 2 = 2$
- The ratio of 8 to 4 is $8 / 4 = 2$
- The ratio of 16 to 8 is $16 / 8 = 2$
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Determine the rule
Since the ratios are consistent, you can conclude that the rule for generating the sequence is multiplying by 2. Thus, the next term can be found by multiplying the last term by 2. -
Formulate the general rule
You can express the general term of the sequence as: $$ a_n = 2^n $$ where ( n ) is the term number, starting from 0.
The rule that generates the sequence is to multiply each term by 2. The general term can be expressed as ( a_n = 2^n ).
More Information
This sequence is an example of a geometric progression, where each term is obtained by multiplying the previous term by a constant factor (in this case, 2). Such sequences appear frequently in various areas of mathematics, including compound interest and population growth.
Tips
- Confusing the rule of addition with multiplication. Always check both differences and ratios to identify the correct operation.
- Assuming the sequence continues in the same manner without verifying the established pattern.
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