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# How to add this ? 1 + 11 + 111 + 9 terms

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Intern
Joined: 25 Mar 2008
Posts: 43
How to add this ? 1 + 11 + 111 + 9 terms [#permalink]

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17 Nov 2008, 03:37
How to add this ?

1 + 11 + 111 + … 9 terms

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SVP
Joined: 17 Jun 2008
Posts: 1504
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 04:00
I will do it as follows:
1 + 11 + 111 + .....up to 9 terms
= 1 + (10 + 1) + (10^2 + 1) + .......
= 9*1 + (10 + 10^2 + ......+ 10^8)
= 9 + 10(10^8-1)/9
Intern
Joined: 25 Mar 2008
Posts: 43
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 04:51
10^2 + 1 =101

but we have 111.
Director
Joined: 14 Aug 2007
Posts: 704
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 06:25
ghentu wrote:
How to add this ?

1 + 11 + 111 + … 9 terms

I am not sure in which format you expect the answer to be, but as long as its 1,11...
answer should be 123456789
Manager
Joined: 08 Aug 2008
Posts: 228
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 06:29
echo alpha_plus_gamma.
if the answer choices is the absolute sum, it is pretty easy to visualise and get the answer as
123456789.
Intern
Joined: 25 Mar 2008
Posts: 43
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 08:15
Yes .....you got the correct answer .

But how did you solve it ?

Current Student
Joined: 28 Dec 2004
Posts: 3311
Location: New York City
Schools: Wharton'11 HBS'12
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 08:22
you have 9 terms, so basically the unit digit willl have a sum of 9, because there 9 1's

ten digit will be 8 because there are 8 1's and so on..

its 9 digit sum where the left most number will be 1..

123456789
Manager
Joined: 08 Aug 2008
Posts: 228
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 08:58
1
KUDOS
ok..for a formulaic approach..here u go...(say the answer is form of $$10^k- xx$$)

$$1+11+111...n$$
$$=1/9[9+99+999...]$$
$$=1/9[(10-1)+(10^2-1)+(10^3-1)...]$$
$$=1/9[10+10^2+10^3....-n]$$
$$=1/9[10(10^n-1)/(10-1) - n]$$

here n=9...
$$=1/9[10/9(10^9-1) -9]$$
SVP
Joined: 17 Jun 2008
Posts: 1504
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 09:48
ghentu wrote:
10^2 + 1 =101

but we have 111.

I think I was feeling sleepy then
Current Student
Joined: 28 Dec 2004
Posts: 3311
Location: New York City
Schools: Wharton'11 HBS'12
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 10:19
prasun84 wrote:
ok..for a formulaic approach..here u go...(say the answer is form of $$10^k- xx$$)

$$1+11+111...n$$
$$=1/9[9+99+999...]$$
$$=1/9[(10-1)+(10^2-1)+(10^3-1)...]$$
$$=1/9[10+10^2+10^3....-n]$$
$$=1/9[10(10^n-1)/(10-1) - n]$$

here n=9...
$$=1/9[10/9(10^9-1) -9]$$

i wish I had known this formula earlier ..
SVP
Joined: 29 Aug 2007
Posts: 2453
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 15:13
fresinha12 wrote:
you have 9 terms, so basically the unit digit willl have a sum of 9, because there 9 1's

ten digit will be 8 because there are 8 1's and so on..

its 9 digit sum where the left most number will be 1..

123456789

I take the above approach to solve this question as it cannot be better than this one..

I agree that the formula approach is universal for ever case but it is not very helpful in this case.
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Intern
Joined: 31 Oct 2008
Posts: 16
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 18:16
Yea, this is probably not meant as a difficult question and it's not realistic to solve that formula during the test within such a short timeframe and without a calculator.

Ghentu, I think what fresinha12 is referring to with visualizing the answer is:

1
11
111
1111
11111
111111
1111111
11111111
+ 111111111
-------------
123456789
Intern
Joined: 25 Mar 2008
Posts: 43
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 18:31
nice explanation.

that helped a lot.

thanks
Manager
Joined: 08 Aug 2008
Posts: 228
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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17 Nov 2008, 22:58
if u notice theres no formula involved except sum of a GP series...trick is to break it down to a 9 series...the approach wud hold good if it was say 7+77+777...or if the answer is of the form a*(10^k+x)...
fresinha12 wrote:
prasun84 wrote:
ok..for a formulaic approach..here u go...(say the answer is form of $$10^k- xx$$)

$$1+11+111...n$$
$$=1/9[9+99+999...]$$
$$=1/9[(10-1)+(10^2-1)+(10^3-1)...]$$
$$=1/9[10+10^2+10^3....-n]$$
$$=1/9[10(10^n-1)/(10-1) - n]$$

here n=9...
$$=1/9[10/9(10^9-1) -9]$$

i wish I had known this formula earlier ..
Director
Joined: 29 Aug 2005
Posts: 836
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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19 Nov 2008, 03:37
prasun84 wrote:
ok..for a formulaic approach..here u go...(say the answer is form of $$10^k- xx$$)

$$1+11+111...n$$
$$=1/9[9+99+999...]$$
$$=1/9[(10-1)+(10^2-1)+(10^3-1)...]$$
$$=1/9[10+10^2+10^3....-n]$$
$$=1/9[10(10^n-1)/(10-1) - n]$$

here n=9...
$$=1/9[10/9(10^9-1) -9]$$

prasun, great effort to explain! +1
One question -
How did you reduce expression in the 4th line to the expression in the 5the line?
Manager
Joined: 08 Aug 2008
Posts: 228
Re: 1 + 11 + 111 + … 9 terms = ? [#permalink]

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19 Nov 2008, 03:59
Sum ofa GP Series...
formula is $$a(r^n-1)/(r-1)$$
where a is 1st term, r is the common ratio..

--== Message from GMAT Club Team ==--

This is not a quality discussion. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Re: 1 + 11 + 111 + … 9 terms = ?   [#permalink] 19 Nov 2008, 03:59
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# How to add this ? 1 + 11 + 111 + 9 terms

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