Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|cite|improve this question
For the record, this was cross-posted on SO: – 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

2 Answers 2

up vote 16 down vote accepted

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.

share|cite|improve this answer
I particularly like this approach since I think it could be use for general problems with repeating digits. – VelvetThunder Dec 19 '10 at 10:31
So if in words how would you to explain/prove it? – Ronnie Dec 20 '10 at 1:25
@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
oh no its not an answer for homework i just needed to calculate the answer, which i did. ( i just noticed a trend and made an answer based on the trend) but if a fellow classmate of mine were to ask me how would i explain it to him instead of saying i just noticed a pattern and took an educated guess – Ronnie Dec 20 '10 at 2:34
@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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.