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What is the 4000th digit following the decimal point in the expansion of 1/176?

 Apr 17, 2022
 #1
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+1

1/176 ==0.00568181818181.............etc.

 

It has a "period" of 2.

 

Therefore: 4000 mod 2==0, which means it should be the 2nd repeating digit, which is "1".

 Apr 17, 2022
 #2
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We can actually divide 1 by 176 and see the result.

 

\(\dfrac1{176} = 0.0056818181\dots= 0.0056\overline{81}\)

 

Therefore, starting from the 5th digit following the decimal point, 

 

\(\text{The }N\text{-th digit following the decimal point} = \begin{cases}8\text{, }N\text{ is odd}\\1, \text{ otherwise}\end{cases}\)

 

Therefore, the 4000th digit is 1.

 Apr 17, 2022

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