# help confirm my answer for the following question

I was given the question:what is 9+99+999+9999+...+999..99(30 digits) After noticing a trend, I came with the conclusion that the answer would be 28 1's 080. Can anyone confirm my answer and give a reason as to why?

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For the record, this was cross-posted on SO: stackoverflow.com/questions/4481022/sum-of-999999-30-9s –  marcog Dec 19 '10 at 0:55
possible duplicate of N= 9 + 99 + 999 + ... 999...99 –  Aryabhata Dec 19 '10 at 1:31
and how would i explain this in words? –  Ronnie Dec 20 '10 at 1:27
@Ronnie: Could you be more specific about what you are having trouble expressing in words? –  Jonas Meyer Dec 20 '10 at 1:43
@Ronnie: It's not; each summand is equal to a power of $10$ minus 1, but the entire sum is not a power of 10 minus 1. In words, you would say what the argument is: Notice that $10^k$ is a $1$ followed by $k$ zeros, so $10^k - 1$ is $k$ 9's. So you can replace each summand with a power of 10 minus 1; then you can reorder the sum so that you add all powers of 10 first, and subtract all the 1s later; then you can figure out what the sum of the powers of 10 is; etc. –  Arturo Magidin Dec 20 '10 at 2:42

Note that $$\underbrace{99\cdots 9}_{k\text{ digits}} = 10^k - 1.$$ So your sum is the same as $$(10-1) + (10^2-1) + (10^3-1) + \cdots + (10^{30}-1),$$ which is equal to $$(10 + 10^2 + 10^3 + \cdots + 10^{30}) - 30.$$ The first sum is easy to do, the difference is easy to do, and it gives your answer.
@Ronnie: How would I explain what? The fact that the number made up of $k$ 9 digits plus 1 equals 10^k? Or what the total of $10 + 10^2 + 10^3 + \cdots + 10^{30}$ is? Or what subtracting 30 form that is? Or why the sum equals $(10+10^2+\cdots+10^{30})-30$? Or what this is equal to? These are pretty simple things; since this is tagged as homework, I really don't think we should be giving you a "ready-to-cut-n-paste-and-turn-in" answer. –  Arturo Magidin Dec 20 '10 at 1:38
@Ronnie: Well, first, that is what you did, so you should say so. Of course, that "educated guess" is not a proof. What I outline above is a proof that your educated guess is correct, after you perform the computation $10+10^2+10^3+\cdots+10^{30}$. –  Arturo Magidin Dec 20 '10 at 2:39
HINT $\$ Exploit the linearity of $\Sigma\:$: $\rm\ \Sigma\ (f(k)+c)\ = \ \Sigma\ f(k) + \Sigma\ c\$ to reduce to a geometric sum.