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# The set X consists of the following terms : {4,44,444,4444,.

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Director
Joined: 29 Nov 2012
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The set X consists of the following terms : {4,44,444,4444,. [#permalink]

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15 Jun 2013, 08:37
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The set X consists of the following terms: {4,44,444,4444,.....), where the nth term has n 4's in it for example 10th term is 4444444444. what is the hundreds digit of the sum of the first 45 terms of set S?

A. 5
B. 3
C. 1
D. 6
E. 9
[Reveal] Spoiler: OA

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Last edited by Bunuel on 15 Jun 2013, 09:20, edited 1 time in total.
Edited the question.
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Re: The set X consists of the following terms : {4,44,444,4444,. [#permalink]

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15 Jun 2013, 08:50
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You can write the sum as $$4(1+11+111+...)$$. Consider now the terms in the brackets: they go from 1 to 1x45 times. Since the question asks for the hundereds digit of the sum you should not bother of any futher digit, so consider the sum of the terms as $$1+11+111+111+111+...$$ , so the sum can be written as $$1+11+(111*43=4773)$$.

As I said before you can discard any digit beyond the hundereds so you have $$4(1+11+773)$$. $$773+1+11=785$$ and $$4*785=3140$$.
$$C)1$$

Hope it's clear
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Re: The set X consists of the following terms : {4,44,444,4444,. [#permalink]

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15 Jun 2013, 09:19
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fozzzy wrote:
The set X consists of the following terms : {4,44,444,4444,.....), where the nth term has n 4's in it for example 10th term is 4444444444. what is the hundreds digit of the sum of the first 45 terms of set S?

A. 5
B. 3
C. 1
D. 6
E. 9

45*4=180, thus the units digit will be 0 and 18 will be carried over to the tens digit;
44*4+18=194, thus the tens digit will be 4 and 19 will be carried over to the hundreds digit;
43*4+19=191, thus the hundreds digit is 1.

Similar question to practice: s-is-the-infinite-sequence-s1-2-s2-22-s3-97477.html

Hope it helps.
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Re: The set X consists of the following terms : {4,44,444,4444,. [#permalink]

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17 Jun 2013, 17:59
Hi Bunuel,

Sorry i could not understand y we r doing this 45*4? 44*4+18? 43*4+19?

Rrsnathan.
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Re: The set X consists of the following terms : {4,44,444,4444,. [#permalink]

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17 Jun 2013, 22:05
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rrsnathan wrote:
Hi Bunuel,

Sorry i could not understand y we r doing this 45*4? 44*4+18? 43*4+19?

Rrsnathan.

Here is how the question works if you write the number down

4444444444 ( 45 4's
444444444 (44 4's
...
..
44444444
4444444
444444
44444
4444
444
44
4 ( 1 4

If you take the sum it will be 45 *4 = 180 the digit will be 0 and there will be a carryover of 18

then to calculate the next set it will be 44*4 + 18 = 194 so this digit is 4 and the carry over is 19

then for the next set it will be 43* 4 + 19 = 191 so this digit is 1, which is the hundreds digit
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Re: The set X consists of the following terms : {4,44,444,4444,. [#permalink]

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17 Jun 2013, 22:55
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rrsnathan wrote:
Hi Bunuel,

Sorry i could not understand y we r doing this 45*4? 44*4+18? 43*4+19?

Rrsnathan.

-----------------------------------------------4
----------------------------------------------44
--------------------------------------------444

...
-- 4,444,444,444,444,444,444,444,444
- 44,444,444,444,444,444,444,444,444
444,444,444,444,444,444,444,444,444

Total 45 numbers.

For the first digit (units place) we should add 45 2's --> 45*4=180, thus the units digit will be 0 and 18 will be carried over to the tens digit;
For the second digit (tens place) we should add 44 4's --> 44*4+18=194, thus the tens digit will be 4 and 19 will be carried over to the hundreds digit;
For the third digit (hundreds place) we should add 43 4's --> 43*4+19=191, thus the hundreds digit is 1.

Similar question to practice: s-is-the-infinite-sequence-s1-2-s2-22-s3-97477.html

Hope it helps.
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The set X consists of the following terms : {4,44,444,4444,. [#permalink]

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08 Dec 2017, 00:29
We only need to be concerned about 45 numbers.
So, in your notebook, write down how the addition would look like, for our set: {4+44+444+...4444}
4
+44
+444
...
Now, you don't really need to write so much, it's just for visualization. So, if there are 45 numbers, we have 45 4's in the Unit's place, 45-1=44 4's in the Ten's place and 44-1=43 4's in the Hundred's place.
So, we need to write down the digits of the sum one by one in the following manner:
45*4=180, write down 0 in the Unit's place, and 18 is carried over.
Then, 44*4=176, so add 18 to 176, we get 194, so write down 4 in the Ten's place and carry over 19.
Then, 43*4=172, so add 19 to 172, we get 191, so write down 1 in the Hundred's place.
We have arrived at our answer choice, C. Do this method on paper for easier understanding. Hope it helps.
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The set X consists of the following terms : {4,44,444,4444,.   [#permalink] 08 Dec 2017, 00:29
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