Mysterious Recurring Decimals

Number Theory Level 3

The decimal representation of the fraction $\frac { 1 }{ 998001 }$ has a very interesting regularity, as shown below:

$\large \dfrac { 1 }{ 998001 } = 0.\, \underbrace{000\ 001\ 002\ 003\ ...\ 999}_{\text{first }\, 3\times 999=2997\, \text{ digits}}\, ...$

If we keep separating the decimal places--from left to right--into a group of 3-digit numbers for a total of 999 times until 999 appears, we have $000,\ 001,\ 002,\ 003,\,...,\ 999.$ But, in fact, the regularity is not perfect as there is one number missing in this sequence. What is the missing number?

Note: If you think it's 007, write 7. If you think it's 077, write 77. If you think it's 777, write 777.

The answer is 998.

1 solution

Rishabh Deep Singh
Jun 1, 2017

Read more about these series Here

Representing the fraction as

$\frac { 1 }{ 998001} =\frac { 1 }{ { 999 }^{ 2 } } =\frac { 1 }{ { 1000 }^{ 2 } } \times \frac { { 1000 }^{ 2 } }{ { \left( 1000-1 \right) }^{ 2 } } =\frac { 1 }{ { 1000 }^{ 2 } } \times \frac { 1 }{ { \left( 1-\frac { 1 }{ 1000 } \right) }^{ 2 } }$

We know that if $|x|<1$

$\frac { 1 }{ (1-x)^{ 2 } } =\displaystyle \sum _{ n=0 }^{ \infty } nx^{ n-1 }=\displaystyle \sum _{ n=1 }^{ \infty } nx^{ n-1 }=1+2x+3{ x }^{ 2 }+4{ x }^{ 3 }+....$

From Taylor Series

if we put $x=\frac { 1 }{ 1000000}$ in the above equation and multiply by $\frac {1}{{ 1000 }^{ 2 }}$ we will get

$\frac { 1 }{ 998001}$ $=$ $\frac { 1 }{ 998001 } =\frac { 1 }{ { 1000 }^{ 2 } } \left[ 1+\frac { 2 }{ { 1000 } } +\frac { 3 }{ { 1000 }^{ 2 } } +\frac { 4 }{ { 1000 }^{ 3 } } +\frac { 5 }{ { 1000 }^{ 4 } } ... \right] \\ =\frac { 1 }{ { 1000 }^{ 2 } } +\frac { 2 }{ { 1000 }^{ 3 } } +\frac { 3 }{ { 1000 }^{ 4 } } +\frac { 4 }{ { 1000 }^{ 5 } } +\frac { 5 }{ { 1000 }^{ 6 } } ...$

$=\quad 0.000001\\ +\quad 0.000000002\\ +...\\ +0.00000....997\\ +0.00000....000998\\ +0.00000....000000999\\ +0.00000....000000001000\\ +.....$

$=\quad 0.000001\\ +\quad 0.000000002\\ +...\\ +0.00000....997\\ +0.00000....000999\\ +0.00000....000000000001\\ +.....$

This all sums up and gives us $0.000001002003.....996997999000001002.....$