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If (-20) + (-19) + (-18) + (-17) + . . .

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If (-20) + (-19) + (-18) + (-17) + . . .

by Brent@GMATPrepNow » Tue May 15, 2018 6:03 am

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If (-20) + (-19) + (-18) + (-17) + . . . + (-2x - 4) = 90, what is the value of x?
A) -24
B) -14
C) -10
D) 10
E) 14

Answer: B
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by Vincen » Wed May 16, 2018 12:44 am
Hello.

I liked this question and I want to try it. <i class="em em-grin"></i>

We have this (-20) + (-19) + (-18) + (-17) + . . . + (-2x - 4) = 90 and we want the value of x.

1. If x>0 then all the numbers in the sum are negative, hence their sum is negative, which is a contradiction because the sum is 90>0. Therefore, we can eliminate the options D and E.

2. Let's try option C, x=-10.

Then we have $$(-20)+(-19)+(-18)+(-17)+...+(-2\left(-10\right)-4)=(-20)+(-19)+(-18)+(-17)+...+(16)$$ $$=\left\{(-20)+(-19)+(-18)+\cdots+\left(-2\right)+\left(-1\right)\right\}+0+\left\{\left(1\right)+\left(2\right)+\cdots+\left(16\right)\right\}$$ $$=(-20)+(-19)+(-18)+\left(-17\right)$$ $$=\ -74\ \ne\ 90.$$ Therefore, we discard the option C.

3. Let's try option A, x=-24.

Then we have $$(-20)+(-19)+(-18)+(-17)+...+(-2\left(-24\right)-4)=(-20)+(-19)+(-18)+(-17)+...+(44)$$ $$=\left\{(-20)+(-19)+(-18)+\cdots+\left(-2\right)+\left(-1\right)\right\}+0+\left\{\left(1\right)+\left(2\right)+\cdots+\left(44\right)\right\}$$ $$=(21)+(22)+(23)++...+(44)$$ $$=\ 780\ \ne\ 90.$$ Therefore, we discard the option A.

4. Finally, let's try option B, x=-14.

Then we have $$(-20)+(-19)+(-18)+(-17)+...+(-2\left(-14\right)-4)=(-20)+(-19)+(-18)+(-17)+...+(24)$$ $$=\left\{(-20)+(-19)+(-18)+\cdots+\left(-2\right)+\left(-1\right)\right\}+0+\left\{\left(1\right)+\left(2\right)+\cdots+\left(24\right)\right\}$$ $$=(21)+(22)+(23)+(24)$$ $$=\ 90 = 90.$$ Therefore, the correct answer is the option [spoiler]B=-14[/spoiler].

I'd like to see a shorter way. <i class="em em-wink"></i>
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by Brent@GMATPrepNow » Wed May 16, 2018 4:47 am
Brent@GMATPrepNow wrote:If (-20) + (-19) + (-18) + (-17) + . . . + (-2x - 4) = 90, what is the value of x?
A) -24
B) -14
C) -10
D) 10
E) 14

Answer: B
Source: www.gmatprepnow.com
Difficulty level: 550-600
First notice that (-20) + (-19) + (-18) + (-17) + . . . + 17 + 18 + 19 + 20 = 0 [because any pair of positive and negative numbers with the same magnitude must add to 0. For example, (-3) + 3 = 0, and (-11) + 11 = 0]

So, we have: (-20) + (-19) + (-18) + (-17) + . . . + 17 + 18 + 19 + 20 + 21 + 22 + ? + ? + ... + (-2x - 4) = 90
Simplify to get: -0 + 21 + 22 + ? + ? + ... + (-2x - 4) = 90

We can see that we won't need to add many consecutive integers (starting at 21) to get to a sum of 90
So, let's just keep adding integers until we reach 90
21 + 22 = 43
21 + 22 + 23 = 66
21 + 22 + 23 + 24 = 90 DONE!

So, it must be the case that -2x - 4 = 24
Add 4 to both sides to get: -2x = 28
Solve: x = -14

Answer: B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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