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

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Joined: 12 Sep 2015
Posts: 4226
If (-20) + (-19) + (-18) + (-17) + . . .  [#permalink]

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15 May 2018, 07:06
Top Contributor
15
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Difficulty:

65% (hard)

Question Stats:

62% (02:15) correct 38% (02:31) wrong based on 125 sessions

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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

*kudos for all correct solutions

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Re: If (-20) + (-19) + (-18) + (-17) + . . .  [#permalink]

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15 May 2018, 08:51
2
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

*kudos for all correct solutions

$$(-20) + (-19) + (-18) + (-17) + . . .0$$ = $$-\frac{20(20+1)}{2}$$

Now we have here : $$-210 + .......... (-2x - 4)$$ = $$90$$

Or, $$0 + ...........(-2x - 4) = 300$$

Now, $$\frac{n ( n + 1 )}{2}$$ = $$300$$

Or, $$\frac{24(24+1)}{2} = 300$$

So, There will be 24 numbers after 0 and the last number will be 24

Or, (-2x - 4) = 24

So, -2x = 28

And $$x = -14$$, IMHO (B)
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If (-20) + (-19) + (-18) + (-17) + . . .  [#permalink]

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15 May 2018, 09:22
3
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

*kudos for all correct solutions

We know that this sequence of consecutive integers starting from -20 and ending at -2x - 4 will give us 90.

Since we have the sum of consecutive integers giving us 90 as a sum and the sequence starts with -20,
the sum of these consecutive integers will be 0 if the sequence ends at 20 as -20 is canceled by 20, 19
cancels -19, 18 cancels -18........., and 1 cancels -1.

Now, the sum of the consecutive integers 21,22,23, and 24 is 90

Since the last number in the series is -2x - 4, $$-2x - 4 = 24$$ -> $$-2x = 28$$

Therefore, the value of x is $$\frac{28}{-2}$$ or -14(Option B)
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Re: If (-20) + (-19) + (-18) + (-17) + . . .  [#permalink]

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16 May 2018, 05:53
Top Contributor
4
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

*kudos for all correct solutions

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

Cheers,
Brent
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Re: If (-20) + (-19) + (-18) + (-17) + . . .  [#permalink]

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11 Jun 2019, 09:52
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Re: If (-20) + (-19) + (-18) + (-17) + . . .   [#permalink] 11 Jun 2019, 09:52
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