I'll Carry You Home

Consider the following series written in form of a decimal A = 1 9 + 1 99 + 1 999 + . . . . + 1 1 0 100 1 A = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + .... + \frac{1}{10^{100} - 1}

Find the digit in the 71 71 st place after the decimal point.


The answer is 3.

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6 solutions

Eddie The Head
Apr 8, 2014

Clearly we have 1 9 = 0.1111111111111111...... \frac{1}{9} = 0.1111111111111111...... 1 99 = 0.010101010101010......... \frac{1}{99} = 0.010101010101010......... 1 999 = 0.00100100100100100100100........ \frac{1}{999} = 0.00100100100100100100100........ So clearly the digit in the k k place will be equal to the number of factors of that number.

But we also have to consider the carry overs..

71 71 is a prime number so without carry over the 71st digit is supposed to be 2 2 .But 72 = 2 3 3 2 72 = 2^{3}*3^{2} has 12 factors!!!And hence will get carried over and added to the 71st digit!!By inspection we can say that there is no significant carry over after the 72 72 nd digit....

Hence the 71 71 st digit of the number should be 2 + 1 = 3 2+1 = \boxed{3}

sir, even though i got it, but your explanation is phew!!!!!!!!!!!

Raghav Gupta - 7 years, 1 month ago

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Do you mean good or bad ?? :p

Eddie The Head - 7 years, 1 month ago

But we start adding from the right hand side,what if there are more carry overs affecting 12 which we got in the 72nd column?

Adarsh Kumar - 7 years, 1 month ago

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That's a good point but here you can intuitively see by inspection that there is no such case....

Eddie The Head - 7 years, 1 month ago

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Yeah and thanks for the reply.

Adarsh Kumar - 7 years, 1 month ago

I forget about that -_-

Figel Ilham - 6 years, 10 months ago

Exactly Same Way.

Kushagra Sahni - 5 years, 5 months ago

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Same way....

Dev Sharma - 5 years, 5 months ago

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Nice Well Done.

Kushagra Sahni - 5 years, 5 months ago

1234567890+×÷=%/*€£@$!#:;&_()-'",.?😰

Kain Hakim - 2 years, 5 months ago

No, 1 99 = 0. 01 \dfrac{1}{99} = 0.\overline{01} , not 0. 10 0.\overline{10} . Similar with 1 999 \dfrac{1}{999} . But the answer is still luckily correct.

Ivan Koswara - 7 years, 2 months ago

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I just made a typographical error which has Nothing to do with the concept....... or the answer for that matter

Eddie The Head - 7 years, 2 months ago

same as mine. Hahahaha

Jayver de Torres - 7 years, 1 month ago
Josh Speckman
Apr 15, 2014

First, let's list given fractions. 1 9 = 0.11111111111 = 0. 1 \dfrac{1}{9} = 0.11111111111 \cdots = 0.\overline{1} . 1 99 = 0.01010101010101 = 0. 01 \dfrac{1}{99} = 0.01010101010101 \cdots = 0.\overline{01} . We find that this pattern continues on to infinity, with 1 999 = 0. 001 \dfrac{1}{999} = 0.\overline{001} , 1 9999 = 0. 0001 \dfrac{1}{9999} = 0.\overline{0001} , etc. Thus there will be 1 1 added to the value of every place by 1 9 \dfrac{1}{9} , another 1 1 added to the value of every other place by 1 99 \dfrac{1}{99} , etc. Since no number less than 100 100 can have a factor greater than 100 100 , and we are working with 71 71 , and 71 < 100 71 < 100 , we don't need to worry about the end fraction. We know that 71 71 has 2 2 factors, and 72 72 has 12 12 . The 1 1 from the 12 12 carries over, and the digit is 3 \fbox{3}

involving the carrys, the 72nd row will have a sum of 12, while the 71st row will have 2, and therefore the digit that will appear at the 71st place will be 2 + 1(carry) = 3

Ramit Das
Apr 17, 2014

nice problem got tricked the first time, overlooked 72 then, later corrected it! Enjoyed solving it!

Bert Seegmiller
Mar 14, 2018

Using the result from a previous quiz problem, counting the decimals ... QED.

0.1223242434262445262644283446282644492448282664303646284844322467482648 3 2246648305432444832.... 0.1223242434262445262644283446282644492448282664303646284844322467482648\boxed{3}2246648305432444832....

Max B
Apr 8, 2014

1/9 is 0.11111111111 1/99 is 0.10101010 1/999 is0.001001001 And so on..... If u notice there is pattern her in recurring of zeros...so 71 th letter has to be 1+1+1.....(understand the pattern)....so 3

Can you explain the pattern? And why is the answer 3?

Calvin Lin Staff - 7 years, 2 months ago

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