The sum of all the integers k such that 26

Question

The sum of all the integers k such that −26 < k < 24 is

(A) 0
(B) −2
(C) −25
(D) −49
(E) −51

ScottTargetTestPrep · Accepted Answer

We must determine the sum of the consecutive integers from -25 to 23, inclusive. To determine the sum we can use the formula sum = average x quantity.
 
To determine quantity, the number of consecutive integers, we compute the following:
 
quantity = largest number – smallest number + 1
 
quantity = 23 – (-25) + 1 = 23 + 25 + 1 = 49
 
Next we must determine the average. Since we have a set of evenly-spaced integers we can determine the average using the formula:
 
average = (largest number + smallest number)/2.
 
average = (-25 + 23)/2 = -2/2 = -1
 
Finally we can determine the sum:
 
sum = quantity x average
 
sum = 49 x -1 = -49.
 
Alternate solution:
 
We must determine the sum of the consecutive integers from -25 to 23 inclusive. However, if we add -23 and 23, the sum will be 0, and so will be the sum of -22 and 22, -21 and 21, and so on. Therefore, the sum of each of these pairs of numbers (as long as they are opposites) is 0. The only numbers left that are not paired with their opposites are -25, -24 and 0. So the sum of all the integers from -25 to 23, inclusive, is the same as the sum of -25, -24 and 0, which is (-25) + (-24) + 0 = -49.
 
Answer: D

HarrishGowtham · Answer

The range of k is from -25 to +23. The range includes 23 pairs of opposite numbers which nullify each other and we are left with just -24 & -25, the sum of which is -49.

Answer choice D

Bunuel · Answer

The sum will be: (-25)+(-24)+(-23)+...+(-1)+0+1+..+23 --> the sum of pairs -23 and 23, -22 and 22 and so on is 0 and we are left only with -25+(-24)=-49.

Or: as we have evenly spaced set: the sum will be average of the first and the last terms multiplied be the # of terms:  \frac{-25+23}{2}*49=-49 .

Answer: D.

Hope it's clear.

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=_IHydI3J5SM

GMATinsight · Answer

The sum of all the integers k such that −26 < k < 24 = SUm of all Integers from -25 to 23

SUM = (-25)+(-24)+------+(23) = (-25)+(-24)+ (-23)+(-22)+------(22)+(23)  = (-49)+ (0)  = -49

Answer: option D

m2k · Answer

Range is given as -26 where -26 and 24 are excluded We can say that -23 to +23 in the range would be cancelled out.. R = {-25,-24,-23 .................+22,+23} => This would give us -25 - 24 = -49.

reto · Answer

Since k defines a range between −26 < k < 24 we can set 0 as the reference point for the negative values and positive values.

The negative values will range from -25 to 0 whereas the positive values will range from 0-23.

We can conclude that for all but -25 and -24 the number pairs will add to 0. So we have left -25 - 24 = -49.

Answer D.

Abhishek009 · Answer

−26 < k < 24

= -25 , -24 ,  -23 ........ k ........... 23  
 
Only -25 & - 24 will remain , all gets cancelled...

Hence answer will be -49

Answer will be (D) - 49

law258 · Answer

Question can be solved using   sum/# = avg   equation:

We know that the total number (#) is 23-(-25)+1 = 49
We also know that since 49 is odd we can pull the 25th number in the sequence and that will be the average

Thus, the equation becomes SUM/49 = -1 --> Manipulating this you will find SUM = -49

Abhishek009 · Answer

−26 < k < 24

So,  −26 < k < 24  =   −25,−24  , −23,−22, −21..................21, 22 , 23

Thus, answer must be (D) −49

TG7 · Answer

We must determine the sum of the consecutive integers from -25 to -1 and from 1 to 23, then we add them together.

-25-24-23-22-21........-1 is -1( 25+24+23+22+21......2+1) or -1(1+2......+21+22+23+24+25)

Now,we have the famous formula for the sum of consecutives integers :n(n+1)/2

So, (1+2......+21+22+23+24+25)= 25(25+1)/2=325

-1* (1+2......+21+22+23+24+25)= -1*  = -325 (1)

The same for:

(1+2.......+20+21+22+23)= 23(23+1)/2=276 (2)

The sum of all the integers k such that −26 < k < 24 is: (1)+ (2)

-1*  +23(23+1)/2= -325+276= -49

Alternate solution:

Sum= Average*Number

average = (largest number + smallest number)/2.
Number= largest number – smallest number + 1

Sum= * = -1*49=-49

dabaobao · Answer

-25, -24, -23 .... 23
-23 to 23 cancel each other.

Remaining: -25, -24

Sum = -49

ANSWER: D

MHIKER · Answer

The number of integers is from -25 to 23 inclusive  = 23-(-25)+1=23+25+1=49

The sum  = \frac{23+(-25)*49}{2}=\frac{-2*49}{2}=-49

The answer is  D

JCGF2021 · Answer

can someone explain me why is -25 the largest number and 23 the smallest number in the average approach. Thank you in advance.

errorlogger · Answer

Hi, the reason why -25 is the larger number is because k > -26. What is the next integer that is greater than -26? 
Similarly, the smaller number is 23 because k < 24. What is the integer that is immediately before 24?

GMATinsight · Answer

Answer: Option D

Video solution by  GMATinsight

https://www.youtube.com/watch?v=wDG-zJB6DxU

100mitra · Answer

Correct Option : D (-49)
-26<K<24
K range is greater than -26 i.e  -25  and less than 24 i.e  23 
 {-23 to 23}  addition is  zero , remains with {-25, -24} i.e (-25)+(-24) = -49

GmatKnightTutor · Answer

Hello, people. Let's get into this.

A quick thing to note is that every negative integer has a positive integer that can cancel it out. For example, a -5 would be cancelled out by a +5.

If we apply this concept here, every integer from -23 to +23 can therefore be ignored because each integer would have an opposite integer within the provided range of k that would cancel it out. The integer 0 is an exception, but because it doesn't affect the sum total we can ignore it as well.

The only integers that do not have an opposite integer within the range are -25 and -24.

The sum total of these two integers is -49.

(D) is your answer.

The sum of all the integers k such that 26 < k < 24 is

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