Note that

where we makes use of the infinite sum of a geometric sequence.

If we convert a fraction into a decimal number, the decimal number is either terminating or recurring. For instance,  is terminating, while  is recurring.

In particular, if the denominator of the fraction is x, then the period of the recurring decimal is at most . For instance, the fraction , when converted into a recurring decimal, is , the period of which is 6.

There is one more interesting issue. Consider the first few digits after the decimal point of the number :

0.010309278350…

You may notice the pattern ‘01’, ‘03’, ‘09’, ‘27’, which are powers of 3. You may wonder why it is so. The reasoning is as follows:

Hence,  is in fact the infinite sum of the geometric sequence with first term 0.01 and common ratio 0.03. This explains the pattern.

Can you work out the first few digits after the decimal point of the fractions  and ?