Question 1197918
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The pattern forming the irrational number 0.120210012000210000120000021... continues indefinitely. 
What is the 589th digit in this pattern?
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        Tutor @math_tutor2020 has great idea about separating the infinite sequence of digits 
        into the sequence of partial strings. But his formula for calculating the total length
        of the union of these strings is not precisely correct.


        So I came to make the necessary corrections. The final conclusion/answer of my analysis
        is the same as tutor @math_tutor2020 has: the 589-th digit of the pattern is 0 (zero).



<pre>
I will interpret your question as if it asks about the 589-th digit after the decimal point.


Our partial strings are 

    S(1) = 120

    S(2) = 2100

    S(3) = 12000

    S(4) = 210000

    S(5) = 1200000

and so on . . . 


We want to get a formula for the length of the union/concatenation  { S(1) U S(2) U S(3) U S(4) U S(5) U . . . U S(n) }.


It is the sum of an arithmetic progression, which can be written formally.

But it can be written in a less formal way from the following considerations: 
each S(k), k = 1, 2, 3, . . .  contains two symbols "1" and "2" in different order 
(which does not matter for us now) and contains k zeroes.  So, the length of the union is

    length { S(1) U S(2) U S(3) U S(4) U S(5) U . . . U S(n) } = (1+2+3+4+5+ . . . +n) + 2n = {{{(n*(n+1))/2}}} + 2n.


             ( * * * it is the place, where I make my correction * * * ).


Now I will use this formula to calculate the length of the union  {U S(n)}  for some values of n that provide 
the length of the union  {U S(n)}  in vicinity of 589.


     n     {U S(n)}

    30       525

    31       558

    32       592

    33       627


From the table, you can see, that the 589-th digit is the digit 3 (three) positions to the left from 
the ending digit of S(32).  Since S(32) has 32-2 = 30 ending zeroes, you conclude that the 589-th digit
of the given pattern after the decimal point is 0 (zero).
</pre>

Solved.