If the sum of consecutive integers from (-20) to (-2x - 4) inclusive

Question

If the sum of consecutive integers from (-20) to (-2x - 4) inclusive is 90, what is the value of x?

A) -24 
B) -14 
C) -10 
D) 10 
E) 14

BrentGMATPrepNow · Accepted Answer

First notice that (-20) + (-19) + (-18) + (-17) + . . . + 17 + 18 + 19 + 20 = 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

Abhishek009 · Answer

(-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)

pushpitkc · Answer

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)

Gemmie · Answer

For total of consecutive numbers from -20 to (-2x-4) to be 90, then it must be that

(-20) + (-19) + ... + (-1) + 0 + 1 + ... + 19 + 20 + 21 + ... + (-2x-4)

-2x-4 > 20
==> 2x + 4 < -20
==> x + 2 < -10
==> x < - 12

So x is either - 14 or - 24

Try x = -14 => -2x - 4 = 28 - 4 = 24

21 + 22 + 23 + 24 = 90

Answer is B

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If the sum of consecutive integers from (-20) to (-2x - 4) inclusive

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