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Decimals

Coins are often used to express a fraction of a currency. In the US, a quarter is $0.25 and a dime is $0.10. Counting change is just one of many ways you can put decimals to use in everyday life.

Level 4

         

In the infinite series

\[ \frac { 1 }{ 9 } +\frac { 1 }{ 99 } +......+\cfrac { 1 }{ { 10 }^{ n }-1 } + \ldots \]

at what place after the decimal point does the third 1 occur?

Note: The first 1 occurs at the first place after the decimal point.


Inspired by this problem

\[0.\overline{0009182736455463728191}=\frac{1}{N}\]

Given that \(N\) is a 4-digit integer, find \(N\).

Note: The repeated digits are from 00, 09, 18, ..., 81,(those multiple 9), followed by 91, the period is 22.

Consider the following series written in form of a decimal \[A = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + .... + \frac{1}{10^{100} - 1}\]

Find the digit in the \(71\)st place after the decimal point.

\[0.\overline{00010203040506\ldots 969799}=\frac{1}{N}\]

Given that \(N\) is a 4-digit integer, find \(N\).

Clarification: The repeated digits are from 00, 01, 02, ..., till 99, without 98, the period is 198.

What is the \(1000^\text{th}\) digit to the right of the decimal point in the decimal representation of \((1+\sqrt{2})^{3000}\)?

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