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Question 1099271: The pattern forming the irrational number 0.543210123450054321000123450000... continues indefinitely. What is the 550th digit in this pattern? Use the leading 5.
Found 2 solutions by Alan3354, greenestamps:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! The pattern forming the irrational number 0.543210123450054321000123450000... continues indefinitely. What is the 550th digit in this pattern? Use the leading 5.
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The repeated sequence has 13 digits.
550/13 = 42.xxxx
13*42 = 546
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The 547th is 5
The 548th is 4
The 549th is 3
The 550th is 2
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The other tutor clearly misunderstood the problem. If the sequence of digits repeated, the number would not be irrational....
The given decimal is
0.54321 0 12345 00 54321 000 12345 0000 54321 00000 12345 000000 ...
I have added spaces to make it easier to analyze the pattern of the digits. If we take each "54321" as the beginning of a new sub-sequence, then...
the first sub-sequence has 13 terms - 2 sequences of 5 nonzero digits, and 1+2=3 zeros
the second sub-sequence has 17 terms - 2 sequences of 5 nonzero digits, and 3+4=7 zeros
the third sub-sequence has 21 terms - 2 sequences of 5 nonzero digits, and 5+6=11 zeros
Clearly the numbers of digits in the successive sub-sequences form the arithmetic sequence
13, 17, 21, 25, ...
To solve the problem, we want to know where the end of one of these sub-sequences is shortly before the 550th term of the given sequence.
The n-th term of the arithmetic sequence 13, 17, 21, 25, ... is  .
The sum of the first n terms of an arithmetic sequence is the average of the first term and last term, multiplied by the number of terms, For this sequence, that is
By one method or another, we find that the largest n for which this is less than 550 is n=14, which gives us a sum of 
That means the 546th term is the last 0 in one of the sub-sequences. The 550th term is 4 digits after that, which is the 4th digit in the "54321" that starts each sub-sequence.
So the 550th digit in the decimal part of the given irrational number is "2".
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