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What is 10 - 8 + 6 - 4 + ... - (-20) ?

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What is 10 - 8 + 6 - 4 + ... - (-20) ? [#permalink] New post 01 Apr 2012, 21:21
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What is 10 - 8 + 6 - 4 + ... - (-20) ?

A. 8
B. 10
C. 12
D. 14
E. 16
[Reveal] Spoiler: OA
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Re: Brute Force or Some Pattern [#permalink] New post 01 Apr 2012, 22:12
Expert's post
GMATPASSION wrote:
What is 10 - 8 + 6 - 4 + ... - (-20) ?

8
10
12
14
16

I solved this by brute force? Any other way.


Pair them up.

10 - 8 = 2
6 - 4 = 2
2 - 0 = 2
.
.
-18 - (-20) = 2

There are a total of 16 terms (2*5 to 2*-10 gives you 5-(-10)+1 = 16 terms) so you will get 8 pairs. Hence the sum will be 2*8 = 16
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Re: Brute Force or Some Pattern [#permalink] New post 01 Apr 2012, 22:18
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Note that 10-8 =2 and 6-4 =2. The next two terms, i.e. 2 and 0 also yield 2. Therefore the series is simply the sum of 2 over many terms.

How many such terms are there? The first negative term is 8 and the last is -20, with a common difference of -4. Alternatively, we could calculate the number of terms by taking the first term as 10, the common difference as -4, and the last term as -18.

-20 = 8 + (n-1) (-4)
=> n = 8

Therefore the answer is simply 2*8 = 16 , or option (E).
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Re: Brute Force or Some Pattern [#permalink] New post 01 Apr 2012, 22:20
Expert's post
GMATPASSION wrote:
What is 10 - 8 + 6 - 4 + ... - (-20) ?

8
10
12
14
16

I solved this by brute force? Any other way.


Or consider them as sum of 2 arithmetic progressions (though it would be much better if you see the pairing pattern)

I) 10 + 6 + 2 + (-2) ....+(-18)

AP with first term = 10 and common diff = -4
Sum = (8/2)(2*10 + 7*(-4)) = -32

II) -8 -4 - 0 - (-4) +..- (-20)
AP with first term = -8 and common diff = 4
Sum = (8/2)(2*(-8) + 7*(4)) = 48

Sum of the series = -32 + 48 = 16
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Re: Brute Force or Some Pattern [#permalink] New post 01 Apr 2012, 22:34
Yep - I was gonna post the same reply below. there are 2 APs here, you can just sum them
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Re: What is 10 - 8 + 6 - 4 + ... - (-20) ? [#permalink] New post 09 Jun 2012, 11:27
some one please enlighten me

please list the whole series and how a common difference of -4 is obtained .

Sorry my limited wisdom is unable to grasp the process

as far as I can see the terms are decreasing by 2 with alternating + and - sign

so the series should continue as 10 -8 +6 - 4 + 2 - 0 + (-2) - ( -4 ) + ( -6) - (-8 ) + ( -10) -( -12) +( -14 )

- (-16) +(-18)- (-20)

10 - 8 + 6 - 4 + 2 - 0 -2 + 4 - 6 + 8 - 10 + 12 - 14 + 16 - 18 + 20


some one please explain how common difference is - 4

please include the complete series so that we can see how this works
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Re: What is 10 - 8 + 6 - 4 + ... - (-20) ? [#permalink] New post 09 Jun 2012, 11:56
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Expert's post
stne wrote:
some one please enlighten me

please list the whole series and how a common difference of -4 is obtained .

Sorry my limited wisdom is unable to grasp the process

as far as I can see the terms are decreasing by 2 with alternating + and - sign

so the series should continue as 10 -8 +6 - 4 + 2 - 0 + (-2) - ( -4 ) + ( -6) - (-8 ) + ( -10) -( -12) +( -14 )

- (-16) +(-18)- (-20)

10 - 8 + 6 - 4 + 2 - 0 -2 + 4 - 6 + 8 - 10 + 12 - 14 + 16 - 18 + 20


some one please explain how common difference is - 4

please include the complete series so that we can see how this works


Here you go:

The sequence is 10-8+6-4+2-0+(-2)-(-4)+(-6)-(-8)+(-10)-(-12)+(-14)-(-16)+(-18)-(-20).

Notice that the odd numbered terms (1st, 3rd, 5th...) form arithmetic progression with common difference of -4 and the even numbered terms (2nd, 4th...) form arithmetic progression with common difference of 4:

The sum of the odd numbered terms is 10+6+2+(-2)+(-6)+(-10)+(-14)+(-18)=10+6+2-2-6-10-14-18=-32;

The sum of the even numbered terms is -8-4-0-(-4)-(-8)-(-12)-(-16)-(-20)=-8-4-0+4+8+12+16+20=48;

Their sum is -32+48=16.

Though I wouldn't recommend to solve this question this way. It's better if you notice that we have 8 pairs:
10-8=2;
6-4=2;
2-0=2;
(-2)-(-4)=2;
(-6)-(-8)=2;
(-10)-(-12)=2;
(-14)-(-16)=2;
(-18)-(-20)=2;

So, the sum of each pair is 2, which makes the whole sum equal to 8*2=16.

Hope it's clear.
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Re: What is 10 - 8 + 6 - 4 + ... - (-20) ? [#permalink] New post 09 Jun 2012, 12:30
Bunuel wrote:
stne wrote:
some one please enlighten me

please list the whole series and how a common difference of -4 is obtained .

Sorry my limited wisdom is unable to grasp the process

as far as I can see the terms are decreasing by 2 with alternating + and - sign

so the series should continue as 10 -8 +6 - 4 + 2 - 0 + (-2) - ( -4 ) + ( -6) - (-8 ) + ( -10) -( -12) +( -14 )

- (-16) +(-18)- (-20)

10 - 8 + 6 - 4 + 2 - 0 -2 + 4 - 6 + 8 - 10 + 12 - 14 + 16 - 18 + 20


some one please explain how common difference is - 4

please include the complete series so that we can see how this works


Here you go:

The sequence is 10-8+6-4+2-0+(-2)-(-4)+(-6)-(-8)+(-10)-(-12)+(-14)-(-16)+(-18)-(-20).

Notice that the odd numbered terms (1st, 3rd, 5th...) form arithmetic progression with common difference of -4 and the even numbered terms (2nd, 4th...) form arithmetic progression with common difference of 4:

The sum of the odd numbered terms is 10+6+2+(-2)+(-6)+(-10)+(-14)+(-18)=10+6+2-2-6-10-14-18=16;

The sum of the even numbered terms is -8-4-0-(-4)-(-8)-(-12)-(-16)-(-20)=-8-4-0+4+8+12+16+20=48;

Their sum is -32+48=16.

Though I wouldn't recommend to solve this question this way. It's better if you notice that we have 8 pairs:
10-8=2;
6-4=2;
2-0=2;
(-2)-(-4)=2;
(-6)-(-8)=2;
(-10)-(-12)=2;
(-14)-(-16)=2;
(-18)-(-20)=2;

So, the sum of each pair is 2, which makes the whole sum equal to 8*2=16.

Hope it's clear.


Thank you , now the previous explanations make sense

Just a small typo in your answer , I think it should be - 32 instead of 16 ( addition of odd terms ) , please do edit , to prevent confusion.

I was not able to see that even and odd terms are making an AP .

as Kudos is a better way of saying thank you , so you have been awarded , will be needing more of your assistance
in the future
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Re: What is 10 - 8 + 6 - 4 + ... - (-20) ? [#permalink] New post 09 Jun 2012, 12:38
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stne wrote:
some one please enlighten me

please list the whole series and how a common difference of -4 is obtained .

Sorry my limited wisdom is unable to grasp the process

as far as I can see the terms are decreasing by 2 with alternating + and - sign

so the series should continue as 10 -8 +6 - 4 + 2 - 0 + (-2) - ( -4 ) + ( -6) - (-8 ) + ( -10) -( -12) +( -14 )

- (-16) +(-18)- (-20)

10 - 8 + 6 - 4 + 2 - 0 -2 + 4 - 6 + 8 - 10 + 12 - 14 + 16 - 18 + 20


some one please explain how common difference is - 4

please include the complete series so that we can see how this works


just rearrange the terms you have written with putting all the terms subtracted together and the ones added together.

10 -(8) +6 -( 4) + 2 - (0) + (-2) - ( -4 ) + ( -6) - (-8 ) + ( -10) -( -12) +( -14 ) - (-16) +(-18)- (-20)

=> 10 +6 + 2 +(-2) +( -6) +(-10)+( -14 ) +(-18) -(8) -( 4) - (0) -( -4 )- (-8 )-( -12) - (-16)- (-20)


Hope that helps.
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Re: What is 10 - 8 + 6 - 4 + ... - (-20) ? [#permalink] New post 17 Jun 2012, 11:52
The quickest strategy for me was to pair them up and add the 2s.

10-8=2;
6-4=2;
2-0=2;
(-2)-(-4)=2;
(-6)-(-8)=2;
(-10)-(-12)=2;
(-14)-(-16)=2;
(-18)-(-20)=2;

2+2+2+2+2+2+2+2= 2*8 = 16
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Re: What is 10 - 8 + 6 - 4 + ... - (-20) ? [#permalink] New post 21 Dec 2012, 06:48
GMATPASSION wrote:
What is 10 - 8 + 6 - 4 + ... - (-20) ?

A. 8
B. 10
C. 12
D. 14
E. 16


I wrote them down and started cancelling pairs (+) against (-) then I was left with 16.

Answer: E
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Re: What is 10 - 8 + 6 - 4 + ... - (-20) ?   [#permalink] 21 Dec 2012, 06:48
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