# Proof that 1/7 is a repeated decimal

This blog post serves as an exercise and solution to the following question:

In plain English: is the fraction

a repeated decimal (0.142857142857142857…)?

## Recurring pattern

How can we tackle such a problem? First note that we have to deal with a recurring pattern: a pattern that refers to itself. One useful concept to model such problems is the concept of sequences. We can model the repeated decimal by the following sequence:

With

How does that work? How does this sequence approximate the repeated fraction? First, take a look at

.

. Now, take a look at the following item in the sequence:

. So, it concatenates “142857” and the recurring pattern constructed so far. By that property we know that the sequence

equals the repeated decimal if

approaches

.

## The proof

We would like to proof that

. This is equal to proving that

. This equals

and this is equal to proving that

.

### Another sequence

Lets introduce another sequence:

Notice that the statement we need to proof is the same as the following:

Now notice that:

Thus:

And thus we know that:

Thus (for

):

From this, we get that:

And from this we can deduce the earlier statement, thus:

Thus,

is indeed equal to the repeated fraction

! If you have any questions or suggestions, feel free to post them below!