The value of a fraction represented in base n is0.111 in over bar while in base 2n it takes the simpler form 0.2n. what is n?

Mathematics
1

the value of a fraction represented in base n is0.111 in over bar while in base 2n it takes the simpler form 0.2n. what is n?

2

the value of a fraction represented in base n is0.111 in over bar while in base 2n it takes the simpler form 0.2n. what is n?

Answer: Let’s first convert the repeating fraction in base n to a simplified fraction:

0.111 in over bar (repeating) in base n can be represented as:

0.111 = 1/n + 1/n^2 + 1/n^3 + …

Using the formula for the sum of an infinite geometric series, we can simplify this:

0.111 = 1/n * (1 / (1 - 1/n)) = 1/n * (n / (n - 1))

Simplifying the expression further:

0.111 = 1 - 1/(n - 1)

Now, let’s consider the simpler form of the fraction in base 2n:

0.2n in base 2n is equivalent to 2n/2n^2 = 1/n

Comparing the simplified fraction from the base n representation with the fraction in base 2n:

1 - 1/(n - 1) = 1/n

This implies that:

1/n = 1/n

This means that the value of the fraction is the same in both base n and base 2n.

Since the fraction remains unchanged when represented in base 2n, and we have the simpler form of 0.2n in base 2n, we can conclude that n must be 2.

So, the value of n is 2.