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On Consecutive & Non-Consecutive Series.

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On Consecutive & Non-Consecutive Series.

by shimmer » Fri Jun 17, 2011 11:54 pm
Hi,

My question is on Consecutive & Non-Consecutive Series. I came by this concept while studying Averages.

Q: What is the sum of all 27 three-digit integers that can be created with the digits 1,2 &3?
Ans: Essentially to find the sum you find the average (avg of lowest value 111 and highest value 333) and you have number of values (27), so now you just find the sum.

I get that part. But my question is about how do i know if these numbers are actually in a consecutive series? Do I need to actually list out all 27 values in this series to make sure they're at equal intervals? It seems a little time consuming + also seems like there's the scope of making a mistake while listing these values out in a hurry.

Is there a quicker approach way to check if the numbers are a Consecutive series? Also, maybe my understanding of what a consecutive series is, is murky. Would love some help with this, too

Thanks!
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by Frankenstein » Sat Jun 18, 2011 2:49 am
Hi,
Let the number be a b c
We know that there are 27 numbers of which 9 numbers end with 1, 9 end with 2 and 9 end with 3
So, the sum of those numbers is 9(1+2+3)
Similarly these 27 numbers have 9 numbers with each of 1,2,3 as 2nd digit.
Sum of the numbers is 9(1+2+3)*10, we multiply with 10 because these digits are in tens place.
Similarly these 27 numbers have 9 numbers with each of 1,2,3 at the hundreds place.
Sum of the numbers is 9(1+2+3)*100, we multiply with 10 because these digits are in hundreds place.

So, sum of number is 9(1+2+3)*(100+10+1) = 111*9*6

Coming to your query of numbers being at equal intervals.
For any number a b c there is a number (4-a) (4-b) (4-c) which is symmetric about 222. So, your formula is justified.
I don't understand the terminology of consecutive and non-consecutive series. So, I cannot comment on that. If you can be more specific, I may explain it.
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by GMATGuruNY » Sat Jun 18, 2011 2:59 am
shimmer wrote:Hi,

My question is on Consecutive & Non-Consecutive Series. I came by this concept while studying Averages.

Q: What is the sum of all 27 three-digit integers that can be created with the digits 1,2 &3?
Ans: Essentially to find the sum you find the average (avg of lowest value 111 and highest value 333) and you have number of values (27), so now you just find the sum.

I get that part. But my question is about how do i know if these numbers are actually in a consecutive series? Do I need to actually list out all 27 values in this series to make sure they're at equal intervals? It seems a little time consuming + also seems like there's the scope of making a mistake while listing these values out in a hurry.

Is there a quicker approach way to check if the numbers are a Consecutive series? Also, maybe my understanding of what a consecutive series is, is murky. Would love some help with this, too

Thanks!
Given a set of numbers that is symmetrical about the median:

Average = (biggest+smallest)/2
Sum = number * average

The set of 3-digit integers that can be formed from the digits 1,2 and 3 is symmetrical about the median (222):

...212,213,221,222,223,231,232...

Thus:
Average = (Biggest+smallest)/2 = (333+111)/2 = 222.
Sum = number * average = 27*222 = 5994.

Another approach:

There are 3 positions: hundreds place, tens place, units place.
There are 3 choices for each position: 1,2, or 3.
Thus, the total number of possible 3-digit integers = 3*3*3 = 27.

Each digit will appear in each position 27/3 = 9 times.
Thus, in each position, there will be nine 1's, nine 2's, and nine 3's.
Sum of the digits in each position = 9*(1+2+3) = 54.

Sum of the hundreds place = 54*100 = 5400.
Sum of the tens place = 54*10 = 540.
Sum of the units place = 54*1 = 54.

Sum of all the integers = 5400+540+54 = 5994.
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by shimmer » Sat Jun 18, 2011 4:17 am
GMATGuruNY wrote:
shimmer wrote:Hi,

My question is on Consecutive & Non-Consecutive Series. I came by this concept while studying Averages.

Q: What is the sum of all 27 three-digit integers that can be created with the digits 1,2 &3?
Ans: Essentially to find the sum you find the average (avg of lowest value 111 and highest value 333) and you have number of values (27), so now you just find the sum.

I get that part. But my question is about how do i know if these numbers are actually in a consecutive series? Do I need to actually list out all 27 values in this series to make sure they're at equal intervals? It seems a little time consuming + also seems like there's the scope of making a mistake while listing these values out in a hurry.

Is there a quicker approach way to check if the numbers are a Consecutive series? Also, maybe my understanding of what a consecutive series is, is murky. Would love some help with this, too

Thanks!
Given a set of numbers that is symmetrical about the median:

Average = (biggest+smallest)/2
Sum = number * average

The set of 3-digit integers that can be formed from the digits 1,2 and 3 is symmetrical about the median (222):

...212,213,221,222,223,231,232...

Thus:
Average = (Biggest+smallest)/2 = (333+111)/2 = 222.
Sum = number * average = 27*222 = 5994.

Another approach:

There are 3 positions: hundreds place, tens place, units place.
There are 3 choices for each position: 1,2, or 3.
Thus, the total number of possible 3-digit integers = 3*3*3 = 27.

Each digit will appear in each position 27/3 = 9 times.
Thus, in each position, there will be nine 1's, nine 2's, and nine 3's.
Sum of the digits in each position = 9*(1+2+3) = 54.

Sum of the hundreds place = 54*100 = 5400.
Sum of the tens place = 54*10 = 540.
Sum of the units place = 54*1 = 54.

Sum of all the integers = 5400+540+54 = 5994.
Thank you. However, when you mention "Given a set of numbers that is symmetrical about the median", how do I gauge that? Specially in this scenario where I know the smallest and the largest value in a jiffy. But knowing the 25 other values in between will take time. Do I have to list these values out in order to check if they're symmetrical about the mean in order to use this approach on the GMAT? Or is there another way?
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by GMATGuruNY » Sat Jun 18, 2011 4:55 am
shimmer wrote:
GMATGuruNY wrote:
shimmer wrote:Hi,

My question is on Consecutive & Non-Consecutive Series. I came by this concept while studying Averages.

Q: What is the sum of all 27 three-digit integers that can be created with the digits 1,2 &3?
Ans: Essentially to find the sum you find the average (avg of lowest value 111 and highest value 333) and you have number of values (27), so now you just find the sum.

I get that part. But my question is about how do i know if these numbers are actually in a consecutive series? Do I need to actually list out all 27 values in this series to make sure they're at equal intervals? It seems a little time consuming + also seems like there's the scope of making a mistake while listing these values out in a hurry.

Is there a quicker approach way to check if the numbers are a Consecutive series? Also, maybe my understanding of what a consecutive series is, is murky. Would love some help with this, too

Thanks!
Given a set of numbers that is symmetrical about the median:

Average = (biggest+smallest)/2
Sum = number * average

The set of 3-digit integers that can be formed from the digits 1,2 and 3 is symmetrical about the median (222):

...212,213,221,222,223,231,232...

Thus:
Average = (Biggest+smallest)/2 = (333+111)/2 = 222.
Sum = number * average = 27*222 = 5994.

Another approach:

There are 3 positions: hundreds place, tens place, units place.
There are 3 choices for each position: 1,2, or 3.
Thus, the total number of possible 3-digit integers = 3*3*3 = 27.

Each digit will appear in each position 27/3 = 9 times.
Thus, in each position, there will be nine 1's, nine 2's, and nine 3's.
Sum of the digits in each position = 9*(1+2+3) = 54.

Sum of the hundreds place = 54*100 = 5400.
Sum of the tens place = 54*10 = 540.
Sum of the units place = 54*1 = 54.

Sum of all the integers = 5400+540+54 = 5994.
Thank you. However, when you mention "Given a set of numbers that is symmetrical about the median", how do I gauge that? Specially in this scenario where I know the smallest and the largest value in a jiffy. But knowing the 25 other values in between will take time. Do I have to list these values out in order to check if they're symmetrical about the mean in order to use this approach on the GMAT? Or is there another way?
If the set of numbers is symmetrical about the median, then the median = the average = (biggest+smallest)/2.

To determine whether a set is symmetrical about the median:

1. Determine the value of (biggest+smallest)/2.
2. Write out a few values to the right and to the left of this result.
3. Check for symmetry.

Given all the 3-digit integers that can be formed from the digits 1, 2 and 3:

1. (biggest+smallest)/2 = (333+111)/2 = 222.
2. Values to the left and right: ...211,212,213...221,222,223...231,231,233...
3. The list above indicates that the set is symmetrical about the median.

To determine the sum of the set, proceed as I did in my post above.
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by amit2k9 » Sat Jun 18, 2011 5:46 am
the 3 digits are abc.
1,2 and 3 will be each repeated 9 times in the series.

thus a*100+b*10+c = 9(1+2+3) * (100+10+1)
= 54*111.

As with the consecutive series mentioned here,you need not list all the numbers here.
(4-a)(4-b)(4-c) will give you the required list though.
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