Hello my name is Leslie, I am trying to help my son with this problem and need help.

In the decimal expansion of 1/17 what digit is in the 1997th place?

many thanks if you can help

Hi Leslie,

I am going to answer the question

In the decimal expansion of ¹/₇ what digit is in the 1997th place?

and you can do ¹/₁₇.

Start by dividing 7 into 1 to put ¹/₇ into decimal form, and look for a pattern.

If you continue, the pattern "142857" repeats.You can see why if you look at the remainders, which I have in red. Once you have a remainder that appears in the list above, the pattern repeats. Here the remainder of 1 results in the repeat. Hence

 ¹/₇ = 0.142857142857142857...

If you continue for 1997 places, how many times does the pattern 142857 repeat? The pattern is 6 characters in length and ¹⁹⁹⁷/₆ = 332 with a remainder of 5. Hence the pattern repeats 332 times and then threre are 5 more digits. They are 14257 and hence the 1997th digit in the decimal representation of ¹/₇ is 7.

Penny

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