Decimals: Level 2 Challenges

         

If the following infinite series SS is evaluated as a decimal,what is the 37th digit to the right of the decimal place?

S=19+199+1999++110n1+ \large S=\frac { 1 }{ 9 } +\frac { 1 }{ 99 } +\frac { 1 }{ 999 } +\ldots + \frac { 1 }{ { 10 }^{ n }-1 } + \ldots

Which of the following is equal to

0.123123123123? 0.123 \, 123 \, 123 \, 123 \, \ldots ?

Without actual division find out which of the following rational numbers has a terminating decimal expansion

P)1790 \frac { 17 }{ 90 }

Q)3350\frac { 33 }{ 50 }

R)12122×53\frac { 121 }{ { 2 }^{ 2 }\times { 5 }^{ 3 } }

S)12123×32×75\frac { 121 }{ { 2 }^{ 3 }\times { 3 }^{ 2 }\times { 7 }^{ 5 } }

True or False:

1>0.99999 1 > 0.99999\ldots

0.2016=0.2016161616=ab0.20\overline{16}=0.2016161616\ldots =\frac{a}{b}

Given that aa and bb are coprime positive integers, find a+ba+b.


Inspiration.

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