Question 190609
# 1 

Q: <font color=red> Use inductive reasoning to determine the next three numbers in the pattern: 1,  1/3,  1/9,  1/27, ... </font>


A:


From the first term 1 to the second term {{{1/3}}}, 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 {{{1/3}}} by {{{3}}}, we get: {{{(1/3)/3=(1/3)(1/3)=1/(3*3)=1/9}}}


So this shows that if we divide {{{1/3}}} by {{{3}}}, we get {{{1/9}}}


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Now let's divide {{{1/9}}}, by {{{3}}} to get:



{{{(1/9)/3=(1/9)(1/3)=1/(9*3)=1/27}}}



So dividing {{{1/9}}} by {{{3}}} gets us {{{1/27}}}



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 {{{1/27}}} by 3 to get 


{{{(1/27)/3=(1/27)(1/3)=1/(27*3)=1/81}}}


So the next term is {{{1/81}}}


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Now divide {{{1/81}}} by 3 to get 


{{{(1/81)/3=(1/81)(1/3)=1/(81*3)=1/243}}}


So the next term is {{{1/243}}}

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So divide {{{1/243}}} by 3 to get 


{{{(1/243)/3=(1/243)(1/3)=1/(243*3)=1/729}}}


So the next term is {{{1/729}}}



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Answer:


So the next three terms of the sequence are {{{1/81}}}, {{{1/243}}}, and {{{1/729}}}