document.write( "Question 1208787: The pattern forming the irrational number 0.3450543003450005430000... continues indefinitely. What is the 81403rd digit in this pattern?
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #847264 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "0.3450543003450005430000\r\n" );
document.write( "\r\n" );
document.write( "I assume you meant digits AFTER the decimal, not including the introductory 0.\r\n" );
document.write( "\r\n" );
document.write( "The 4's occur in positions 2, 6, 11, 17, ...\r\n" );
document.write( "\r\n" );
document.write( "Its sequence of first differences 4, 5, 6,... is linear, so the\r\n" );
document.write( "sequence of positions of 4s is quadratic.\r\n" );
document.write( "\r\n" );
document.write( "Also if 4 has occurred an even-number of times, the digit that follows it is a\r\n" );
document.write( "3, and if 4 has occurred an odd-number of times, the digit that follows it is a\r\n" );
document.write( "5.\r\n" );
document.write( "\r\n" );
document.write( "Let the general term of 2, 6, 11, 17, ... be \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Solve that and get A=0.5, B=2.5, C=-1\r\n" );
document.write( "\r\n" );
document.write( "So the general term for the positions of 4's is \r\n" );
document.write( "\r\n" );
document.write( "So let's find the position of the 4 nearest the 81403rd term by setting\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Solving that with a quadratic program on my TI-84 graphing calculator:\r\n" );
document.write( "\r\n" );
document.write( "n=401.0024783 and n=-406.0024783\r\n" );
document.write( "\r\n" );
document.write( "So it is almost equivalent to a quadratic factorable as (n-401)(n+406) which\r\n" );
document.write( "would have been this quadratic in n:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The first two terms of that are twice the first two terms of the general\r\n" );
document.write( "equation for the positions of the 4's. So we divide through by 2\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Adding 81403 to both sides\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Adding -1 makes the left side the general term for digit positions of \r\n" );
document.write( "the 4's. So let's add -1 to the right side also:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "That has solutions 401 and -406, so the 81402nd digit is a 4 and since \r\n" );
document.write( "it is the 401st occurrence of a 4, and 401 is odd, the next digit, or the\r\n" );
document.write( "81403rd digit, is a 5.\r\n" );
document.write( "\r\n" );
document.write( "BTW, if you did include the introductory 0 left of the decimal point, then the\r\n" );
document.write( "answer would have been 4, and the solution would have been a little easier.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );