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! :)
(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.
