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# 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 =

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Re: 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = [#permalink]
3
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GMATPrepNow wrote:
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

-220
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Re: 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = [#permalink]
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GMATPrepNow wrote:
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

$$4+36+100+14^2+18^2-16-64-144-256-400$$

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

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

ANS : C
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Re: 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = [#permalink]
1
Kudos
GMATPrepNow wrote:
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

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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Re: 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = [#permalink]
4
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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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Re: 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = [#permalink]
2
Kudos
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 = [#permalink]
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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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Re: 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = [#permalink]
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 = [#permalink]
1
Kudos
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 = [#permalink]
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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Re: 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = [#permalink]
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Re: 2^2 - 4^2 + 6^2 - 8^2 + 10^2 - 12^2 + 14^2 - 16^2 + 18^2 - 20^2 = [#permalink]
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