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Let S be the set of all real numbers of the form

(a_1)/(3) + (a_2)/(3^2) + (a_3)/(3^3) + ...

where (a_i) is equal to either 1 or -1 for each i.

(a) Is the number 1/4 in the set S? 

(b) Is the number 1/7 in the set S?

Please help and include an explanation! :) 

 May 5, 2023
 #1
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(a) No, 1/4 is not in the set S.

We can write 1/4 as follows:

1/4=1/3​+1/3​

Since 1 and -1 are the only possible values for ai​, we can see that the only way to represent 1/4 as a sum of the form a1/3​​+a2/3^2​​+a3/3^3​​+... is to have a1​=a2​=1. However, this is not possible, since ai​ can only be 1 or -1. Therefore, 1/4 is not in the set S.

(b) Yes, 1/7 is in the set S.

We can write 1/7 as follows:

1/7 = 1/3 - 2/3^2

Since 1 and -1 are the only possible values for ai​, we can see that the only way to represent 1/7 as a sum of the form a1/3​​+a2​/3^2​+a33^2​​+... is to have a1​=1 and a2​=−2. This is possible, since ai​ can be 1 or -1. Therefore, 1/7 is in the set S.

 May 5, 2023
 #2
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How does 1/4 =1/3+1/3, and how does 1/7= 1/3- 2/3^2?

Guest May 5, 2023

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