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555-605 (Medium)|   Algebra|   Arithmetic|   Sequences|      
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BrentGMATPrepNow
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = Updated on: 02 May 2017, 10:15
Originally posted by BrentGMATPrepNow on 02 May 2017, 06:44.
Last edited by Bunuel on 02 May 2017, 10:15, edited 1 time in total.
Edited the question.
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

35% (medium)

Question Stats:

77% (02:29) correct 23% (02:51) wrong based on 691 sessions

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\(2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 =\)

A) -40
B) -110
C) -220
D) -440
E) -880

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C
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BrentGMATPrepNow
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 02 May 2017, 11:37
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GMATPrepNow
\(2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 =\)

A) -40
B) -110
C) -220
D) -440
E) -880

Show more

Here's a different approach:

2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² = (2² - 4²) + (6² - 8²) + (10² - 12²) + (14² - 16²) + (18² - 20²)
= (2 - 4)(2 + 4) + (6 - 8)(6 + 8) + (10 - 12)(10 + 12) + (14 - 16)(14 + 16) + (18 - 20)(18 + 20)
= (-2)(2 + 4) + (-2)(6 + 8) + (-2)(10 + 12) + (-2)(14 + 16) + (-2)(18 + 20)
= -2(2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20)
= -2(2)(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
= (-4)(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
= (-4)(55)
= -220

Answer:
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C

Cheers,
Brent
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 02 May 2017, 07:05
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GMATPrepNow
2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

A) -40
B) -110
C) -220
D) -440
E) -880

*kudos for all correct solutions
Show more

2² - 4²=-12
6² - 8²=-28
10² - 12²=-44
14² - 16² =-60
18² - 20²=-76

We can see the pattern here, adding all we get -220

Ans:C

Method 2:
(2² + 6² + 10² + 14² + 18² )-( 4² + 8² + 12²+ 16² + 20²)=-220
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 02 May 2017, 06:58
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GMATPrepNow
2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

A) -40
B) -110
C) -220
D) -440
E) -880

*kudos for all correct solutions
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-220
Working attached..
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 02 May 2017, 07:06
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GMATPrepNow
2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

A) -40
B) -110
C) -220
D) -440
E) -880

*kudos for all correct solutions
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\(4+36+100+14^2+18^2-16-64-144-256-400\)

\(140+(7*2)^2+(9*2)^2-880\)

\(140+(49*4)+(81*4)-880=-220\)

ANS : C
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 02 May 2017, 08:58
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GMATPrepNow
2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

A) -40
B) -110
C) -220
D) -440
E) -880

*kudos for all correct solutions
Show more

2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

Or, \(4 - 16 + 36 - 64 + 100 - 144 + 196 - 256 + 324 - 400\)

Or, \(( 4 + 36 + 100 + 196 + 324 ) - ( 16 + 64 + 144 + 256 + 400 )\)

Or, \(660 - 880 = - 220\)

Hence, answer must be (C) - 220
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 02 May 2017, 10:07
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factor out 2^2

2^2(1- 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + 7^2 - 8^2 + 9^2 - 10^2)=
= 4*(-3 -7 -11 -15 -19)
=-220
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 02 May 2017, 15:03
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Separate each of the a^2-b^2 terms as (a-b)(a+b)

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

All the (a-b) terms are equal to -2. Therefore we can use

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

=-4(1+2+3+4+5+6+7+8+9+10)

Sum of n consecutive integers = n(n+1)/2. Therefore, term in the parantheses is 10x11/2 = 55

Therefore answer = -4x55 = -220. Option (c)
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 09 May 2017, 22:19
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I used the following method:

2^2-4^2= (2-4)(2+4)= -12
6^2-8^2= (6-8)(6+8)= -28
10^2-12^2= (10-12)(10+12)= -44

Common Difference (d) = -16

so, 5/2 [ 2 x-12 + (5-1) x-16]
> 5/2 [ -24 -64]
> 5/2 x -88
> 5 x -44 => -220
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 15 May 2017, 09:50
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2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

4−16+36−64+100−144+196−256+324−4004−16+36−64+100−144+196−256+324−400

(4+36+100+196+324)−(16+64+144+256+400)(4+36+100+196+324)−(16+64+144+256+400)

660−880=−220
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 07 Nov 2019, 04:16
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The above solutions are way too complicated

the first difference is \(-12\)
the second difference is \(-28=-12-16\)
the third difference is \(-44=-28-16\)

Now we see that the common difference is \(-16\), hence \(5*-12+16*-10\) is the answer
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 10 Apr 2021, 13:25
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Looking at the terms, it makes more sense to pair the square terms in the following order:

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

= (2+12)(2-12)+(14+4)(14-4)+(6+16)(6-16)+(18+8)(18-8)+(10+20)(10-20)
= (14)(-10)+(18)(10)+(22)(-10)+(26)(10)+(30)(-10)
= -140+180-220+260-300
= -220
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2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = 12 Nov 2025, 20:39
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