document.write( "Question 1197918: The pattern forming the irrational number 0.120210012000210000120000021... continues indefinitely. What is the 589th digit in this pattern? \n" ); document.write( "

Algebra.Com's Answer #831402 by ikleyn(52814) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "The pattern forming the irrational number 0.120210012000210000120000021... continues indefinitely.
\n" ); document.write( "What is the 589th digit in this pattern?
\n" ); document.write( "~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        Tutor @math_tutor2020 has great idea about separating the infinite sequence of digits
\n" ); document.write( "        into the sequence of partial strings. But his formula for calculating the total length
\n" ); document.write( "        of the union of these strings is not precisely correct.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        So I came to make the necessary corrections. The final conclusion/answer of my analysis
\n" ); document.write( "        is the same as tutor @math_tutor2020 has: the 589-th digit of the pattern is 0 (zero).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "I will interpret your question as if it asks about the 589-th digit after the decimal point.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Our partial strings are \r\n" );
document.write( "\r\n" );
document.write( "    S(1) = 120\r\n" );
document.write( "\r\n" );
document.write( "    S(2) = 2100\r\n" );
document.write( "\r\n" );
document.write( "    S(3) = 12000\r\n" );
document.write( "\r\n" );
document.write( "    S(4) = 210000\r\n" );
document.write( "\r\n" );
document.write( "    S(5) = 1200000\r\n" );
document.write( "\r\n" );
document.write( "and so on . . . \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "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) }.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is the sum of an arithmetic progression, which can be written formally.\r\n" );
document.write( "\r\n" );
document.write( "But it can be written in a less formal way from the following considerations: \r\n" );
document.write( "each S(k), k = 1, 2, 3, . . .  contains two symbols \"1\" and \"2\" in different order \r\n" );
document.write( "(which does not matter for us now) and contains k zeroes.  So, the length of the union is\r\n" );
document.write( "\r\n" );
document.write( "    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 =  + 2n.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "             ( * * * it is the place, where I make my correction * * * ).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now I will use this formula to calculate the length of the union  {U S(n)}  for some values of n that provide \r\n" );
document.write( "the length of the union  {U S(n)}  in vicinity of 589.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     n     {U S(n)}\r\n" );
document.write( "\r\n" );
document.write( "    30       525\r\n" );
document.write( "\r\n" );
document.write( "    31       558\r\n" );
document.write( "\r\n" );
document.write( "    32       592\r\n" );
document.write( "\r\n" );
document.write( "    33       627\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the table, you can see, that the 589-th digit is the digit 3 (three) positions to the left from \r\n" );
document.write( "the ending digit of S(32).  Since S(32) has 32-2 = 30 ending zeroes, you conclude that the 589-th digit\r\n" );
document.write( "of the given pattern after the decimal point is 0 (zero).\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );