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Suppose $N = 1+11+101+1001+10001+\dots+1\underbrace{00\dots00}_{50\text{ zeroes}}1$.

When $N$ is written as a single integer, i.e. all terms added, what is the sum of the digits of $N$?

I tried subtracting $1$ from each term to get:

$$0+10+100+\dots\;,$$

therefore ending with a sum of $\underbrace{111\dots111}_{50\text{ ones}}$.

Then I added $50$ to the end two numbers of $N$ ($50$ is the total number of ones I minused); there the end two numbers would be $6$ and $0$; therefore add all the ones, ($49$ I think) and the $6$ to obtain $55$.

Not sure if this is the way to do it though?

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I may have miscounted my ones –  Daniel May 15 '12 at 7:45
    
The method seems all right. Be careful though in counting things; for instance your sum has $52$ terms to begin with. –  Marc van Leeuwen May 15 '12 at 7:46
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1 Answer 1

up vote 3 down vote accepted

You might try again to count the numbers of terms. If you had started with $1+11+101 =113$ your method seems to give an answer of $2$ when the answer should be $5$.

I think you have $52$ terms in the original sum, giving $51$ ones in $111\ldots1110$ to which you add $52$, giving giving $50$ ones in $111\ldots1162$ a digit sum of $58$.

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do you add 52 though? because we subtracted the ones, we must add them at the end of the number not the actual number 52 –  Daniel May 15 '12 at 7:47
    
@Daniel: yes - as in "to which you add $52$" –  Henry May 15 '12 at 7:48
    
on the end we have 1110 i propose we add the 52 to the 10 at the end to obtain 62 at the end –  Daniel May 15 '12 at 7:48
    
no you add the sum of the digits not the digit itself –  Daniel May 15 '12 at 7:48
    
@Daniel: I have adjusted my reply –  Henry May 15 '12 at 7:51
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