SOLUTION: # 1. Use inductive reasoning to determine the next three numbers in the pattern: 1, 1/3, 1/9, 1/27, ...

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Question 190609: # 1. Use inductive reasoning to determine the next three numbers in the pattern: 1, 1/3, 1/9, 1/27, ...
Answer by jim_thompson5910(35256) About Me  (Show Source):
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# 1
Q: Use inductive reasoning to determine the next three numbers in the pattern: 1, 1/3, 1/9, 1/27, ...

A:

From the first term 1 to the second term 1%2F3, notice how the second term is just the first term divided by 3. So let's see if this holds from the second term to the third term.


If we divide 1%2F3 by 3, we get: %281%2F3%29%2F3=%281%2F3%29%281%2F3%29=1%2F%283%2A3%29=1%2F9

So this shows that if we divide 1%2F3 by 3, we get 1%2F9

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Now let's divide 1%2F9, by 3 to get:


%281%2F9%29%2F3=%281%2F9%29%281%2F3%29=1%2F%289%2A3%29=1%2F27


So dividing 1%2F9 by 3 gets us 1%2F27


So using inductive reasoning, we would conjecture that this pattern continues indefinitely.

So this means that we can find the next three terms by dividing each previous term by 3.

So divide 1%2F27 by 3 to get

%281%2F27%29%2F3=%281%2F27%29%281%2F3%29=1%2F%2827%2A3%29=1%2F81

So the next term is 1%2F81

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Now divide 1%2F81 by 3 to get

%281%2F81%29%2F3=%281%2F81%29%281%2F3%29=1%2F%2881%2A3%29=1%2F243

So the next term is 1%2F243
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So divide 1%2F243 by 3 to get

%281%2F243%29%2F3=%281%2F243%29%281%2F3%29=1%2F%28243%2A3%29=1%2F729

So the next term is 1%2F729


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Answer:

So the next three terms of the sequence are 1%2F81, 1%2F243, and 1%2F729