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Somewhat tricky calculation: 2² - 4² + 6² - 8² + ...

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by Brent@GMATPrepNow » Tue May 02, 2017 5:46 am
2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

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

Answer: C

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by DavidG@VeritasPrep » Tue May 02, 2017 6:37 am
Brent@GMATPrepNow wrote:2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

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

Answer: C

Source: www.gmatprepnow.com
Difficulty level: 650 -700
Fun one. If we group the terms into pairs, a pattern emerges: 2² - 4² = -12; 6² - 8² = -28; 10² - 12² = -44
Each successive term is 16 less than the previous term. We'll have 5 pairs, so 5 terms: -12, -28, -44, -60, -76. We could just add them here and get -220 or C
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by DavidG@VeritasPrep » Tue May 02, 2017 6:39 am
Time Saver:

In an evenly spaced set, the mean = median = (high + low)/2
First term = -12
Last term = -76

Mean (average) = (-12+-76)/2 = -44
There are 5 terms.
Average * # terms = (-44) * 5 = -220
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by Brent@GMATPrepNow » Tue May 02, 2017 10:38 am
Brent@GMATPrepNow wrote:2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

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

Answer: C

Source: www.gmatprepnow.com
Difficulty level: 650 -700
Another 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)
= [spoiler]-220
[/spoiler]
Answer: C

Cheers,
Brent
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by DavidG@VeritasPrep » Wed May 03, 2017 8:38 am
Brent@GMATPrepNow wrote:
Brent@GMATPrepNow wrote:2² - 4² + 6² - 8² + 10² - 12² + 14² - 16² + 18² - 20² =

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

Answer: C

Source: www.gmatprepnow.com
Difficulty level: 650 -700
Another 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)
= [spoiler]-220
[/spoiler]
Answer: C

Cheers,
Brent
Nice difference of squares action there. Elegant solution, Brent.
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