Download Problem of the Week Problem D and Solution A Repeat Performance

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Problem of the Week
Problem D and Solution
A Repeat Performance
Problem
1
is written as a decimal, what digit occurs in the 2014th
700 000 000
place after the decimal point?
When
Solution
Notice that
1
1
1
1
=
× = 0.000 000 01 × .
700 000 000 100 000 000 7
7
Also, note that 17 = 0.142857. That is, when 17 is written as a decimal, the
digits after the decimal point occur in repeating blocks of the 6 digits 142857.
Therefore,
1
1
= 0.000 000 01× = 0.000 000 01×0.142 857 = 0.000 000 001 428 57.
700 000 000
7
That is, when
1
700 000 000
is written as a decimal, the digits after the decimal
point will be eight 0’s followed by repeating blocks of the 6 digits 142 857.
1
We see the decimal representation of 700 000
000 has the same repetition as that
for 17 , but the pattern is shifted over 8 places. Therefore, the 2014th digit after
1
the decimal point when 700 000
000 is written as a decimal is the same as the
(2014 − 8) = 2006th digit after the decimal point when 17 is written as a
decimal.
2006
1
= 334 , then the 2006th digit after the decimal point occurs after
6
3
334 blocks of the repeating digits have been used. In 334 blocks of 6 digits,
there are 334 × 6 = 2004 digits in total.
Therefore, the 2006th digit is 2 digits into the 335th block, so must be 4.
Since
Therefore, the 2014th digit after the decimal point in the decimal
1
representation of 700 000
000 is a 4.