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# If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +

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CEO
Joined: 12 Sep 2015
Posts: 2587
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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08 May 2017, 07:06
Top Contributor
13
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Difficulty:

95% (hard)

Question Stats:

42% (02:08) correct 58% (01:44) wrong based on 217 sessions

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If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

*kudos for all correct solutions

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Brent Hanneson – Founder of gmatprepnow.com

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Joined: 05 Mar 2015
Posts: 978
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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08 May 2017, 07:53
3
1
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

*kudos for all correct solutions

simply put j= 2+4
k= 1+3

then 1^2- 2^2 + 3^2 - 4^2 = 1-4+9-16= -10

also j= 2+4= 6 && k = 1+3 =4
j^2= 36 && k^2 =16

just plug in values to option to get -10 as our answer

only option C does

Ans C
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Joined: 02 Aug 2009
Posts: 5938
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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08 May 2017, 08:25
6
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

*kudos for all correct solutions

Hi,

$$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$ .....
Here take all inpairs $$1^2-2^2$$, $$3^2-4^2$$, and so on till $$99^2-100^2$$..
1^2-2^2=(1-2)(1+2)=-1(1+2)=-1-2...
3^2-4^2=(3-4)(3+4)=-1(3+4)=-3-4..
So the equation becomes -1-2-3-4-......-99-100=-(1+2+3+4+...+99+100)= -[(2+4+6....+98+100)+(1+3+5+...+97+99)]=-[(j)+(k)]=-j-k
C
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Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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08 May 2017, 10:52
1
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

*kudos for all correct solutions

We can break the problem into a^2 - b ^2 = 1^2 - 2^2 = (1+2) (1-2) = -3
similarly other pair will give = -7 ,next pair will give = -11
final pair will give -199
Now the question stem is reduced to below seq:
-3 -7-11.....-199

-3 = -(1+2)
-7 = -(3+4)
-11 = -(5+6)

-(1+3+5....)-(2+4+6...)
-(j)-(k)
-(j+k)..Ans
CEO
Joined: 12 Sep 2015
Posts: 2587
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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10 May 2017, 14:07
8
Top Contributor
2
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

We have several differences of squares hiding in the expression 1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100²

1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100² = 1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100²
= (1 - 2)(1 + 2) + (3 - 4)(3 + 4) + (5 - 6)(5 + 6) + . . . . . + (97 - 98)(97 + 98) + (99 - 100)(99 + 100)
= (-1)(1 + 2) + (-1)(3 + 4) + (-1)(5 + 6) + . . . . . + (-1)(97 + 98) + (-1)(99 + 100)
= (-1)[(1 + 2) + (3 + 4) + (5 + 6) + . . . . . + (97 + 98) + (99 + 100)]
= (-1)(1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100)

IMPORTANT: within the sum, 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100, we have all of the ODD integers from 1 to 99 inclusive, and we have all of the EVEN integers from 2 to 100 inclusive.

So, we can say that 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100 = K + J

So, we're replace 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100 with K + J.
We get: (-1)(1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100) = (-1)(K + J)
= -K - J

Cheers,
Brent
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If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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11 May 2017, 22:52
GMATPrepNow wrote:
If $$J = 2 + 4 + 6 + 8 + . . . 98 + 100$$, and $$K = 1 + 3 + 5 + 7 + . . . + 97 + 99$$, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

*kudos for all correct solutions

From Matt (Veritas Prep)

We could also cheat with a pattern:

$$n² - (n + 1)² => n² - (n² + 2n + 1) => -(2n + 1) => -(n + n + 1)$$ for any value of n.

Since we've got 1² - 2² + 3² - 4² ..., we've really got -(1 + 2) -(3 + 4) .... -(99 + 100), or -1 -2 -3 -4 .... - 99 - 100, or -(1 + 2 + 3 + ... + 100), or -(K + J), or -K - J.
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If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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11 May 2017, 23:28
1
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

*kudos for all correct solutions

There can be many solutions possible. But we will go with the basic solution, though it might be lengthy one to understand the concepts..
We can use tricks to solve problem as suggested by other members in exam.. Learning tricks is also very important.

J = 2+4+6+8+....+98+100
K = 1+3+5+7+....++97+99

$$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$
= (1-2)(1+2) + (3-4) (3+4)+ (5-6)(5+6) +.......+ (97-98)(97+98) +(99-100)(99+100)
= -1[(1+2)+(3+4)(5+6) +......+ (97+98)+(99+100)]
= -1[ (1+3+5+...+97+99) + (2+4+6+...+98+100)]
= -1(K+J)
= -K-J

Anwer : C

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Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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11 May 2017, 23:39
1
We can pair any 2 consecutive term and apply (a+b) (a-b) in stead of a^2 - b ^2.

Sent from my Moto G (4) using GMAT Club Forum mobile app
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Joined: 05 Mar 2018
Posts: 3
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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30 Apr 2018, 21:01
GMATPrepNow wrote:
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

We have several differences of squares hiding in the expression 1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100²

1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100² = 1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100²
= (1 - 2)(1 + 2) + (3 - 4)(3 + 4) + (5 - 6)(5 + 6) + . . . . . + (97 - 98)(97 + 98) + (99 - 100)(99 + 100)
= [color=red](-1)(1 + 2) + (-1)(3 + 4) + (-1)(5 + 6) + . . . . . + (-1)(97 + 98) + (-1)(99 + 100)
= (-1)[(1 + 2) + (3 + 4) + (5 + 6) + . . . . . + (97 + 98) + (99 + 100)]
= (-1)(1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100)

IMPORTANT: within the sum, 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100, we have all of the ODD integers from 1 to 99 inclusive, and we have all of the EVEN integers from 2 to 100 inclusive.

So, we can say that 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100 = K + J

So, we're replace 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100 with K + J.
We get: (-1)(1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100) = (-1)(K + J)
= -K - J

Cheers,
Brent

Hi Brent, I did't understand the step 3. Can you please elaborate, how did you write
(1-2)(1+2) = (-1)(1 + 2)
(3-4)(3+4) =(-1)(3+4)... and so on...?
CEO
Joined: 12 Sep 2015
Posts: 2587
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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01 May 2018, 08:16
2
Top Contributor
MayurAgrawal wrote:
GMATPrepNow wrote:
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then $$1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =$$

A) J² - K²
B) -50(J² - K²)
C) -K - J
D) K² - J²
E) (-J - K)²

We have several differences of squares hiding in the expression 1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100²

1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100² = 1² - 2² + 3² - 4² + 5² - 6² + . . . . . + 97² - 98² + 99² - 100²
= (1 - 2)(1 + 2) + (3 - 4)(3 + 4) + (5 - 6)(5 + 6) + . . . . . + (97 - 98)(97 + 98) + (99 - 100)(99 + 100)
= (-1)(1 + 2) + (-1)(3 + 4) + (-1)(5 + 6) + . . . . . + (-1)(97 + 98) + (-1)(99 + 100)
= (-1)[(1 + 2) + (3 + 4) + (5 + 6) + . . . . . + (97 + 98) + (99 + 100)]
= (-1)(1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100)

IMPORTANT: within the sum, 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100, we have all of the ODD integers from 1 to 99 inclusive, and we have all of the EVEN integers from 2 to 100 inclusive.

So, we can say that 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100 = K + J

So, we're replace 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100 with K + J.
We get: (-1)(1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100) = (-1)(K + J)
= -K - J

Cheers,
Brent

Hi Brent, I did't understand the step 3. Can you please elaborate, how did you write
(1-2)(1+2) = (-1)(1 + 2)
(3-4)(3+4) =(-1)(3+4)... and so on...?

Sure thing.
Let's take (1 - 2)(1 + 2) + (3 - 4)(3 + 4) + (5 - 6)(5 + 6) + . . . . . + (97 - 98)(97 + 98) + (99 - 100)(99 + 100) and break it into its individual parts:
(1 - 2)(1 + 2) = (-1)(1 + 2) because 1 - 2 = -1
(3 - 4)(3 + 4) = (-1)(3 + 4) because 3 - 4 = -1
(5 - 6)(5 + 6) = (-1)(5 + 6) because 5 - 6 = -1
.
.
.

(97 - 98)(97 + 98) = (-1)(97 + 98) because 97 - 98 = -1
(99 - 100)(99 + 100) = (-1)(99 + 100) because 99 - 100 = -1

So, we get: (-1)(1 + 2) + (-1)(3 + 4) + (-1)(5 + 6) + . . . . . + (-1)(97 + 98) + (-1)(99 + 100)
From here, we can factor out the -1 to get: (-1)[(1 + 2) + (3 + 4) + (5 + 6) + . . . . . + (97 + 98) + (99 + 100)]
Which is the same as: (-1)(1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100)

Does that help?

Cheers,
Brent
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Intern
Joined: 05 Mar 2018
Posts: 3
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + [#permalink]

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01 May 2018, 18:00
Quote:
Quote:
Hi Brent, I did't understand the step 3. Can you please elaborate, how did you write
(1-2)(1+2) = (-1)(1 + 2)
(3-4)(3+4) =(-1)(3+4)... and so on...?

Sure thing.
Let's take (1 - 2)(1 + 2) + (3 - 4)(3 + 4) + (5 - 6)(5 + 6) + . . . . . + (97 - 98)(97 + 98) + (99 - 100)(99 + 100) and break it into its individual parts:
(1 - 2)(1 + 2) = (-1)(1 + 2) because 1 - 2 = -1
(3 - 4)(3 + 4) = (-1)(3 + 4) because 3 - 4 = -1
(5 - 6)(5 + 6) = (-1)(5 + 6) because 5 - 6 = -1
.
.
.

(97 - 98)(97 + 98) = (-1)(97 + 98) because 97 - 98 = -1
(99 - 100)(99 + 100) = (-1)(99 + 100) because 99 - 100 = -1

So, we get: (-1)(1 + 2) + (-1)(3 + 4) + (-1)(5 + 6) + . . . . . + (-1)(97 + 98) + (-1)(99 + 100)
From here, we can factor out the -1 to get: (-1)[(1 + 2) + (3 + 4) + (5 + 6) + . . . . . + (97 + 98) + (99 + 100)]
Which is the same as: (-1)(1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100)

Does that help?

Cheers,
Brent

Silly Me. I totally missed subtraction. Thank you very much for clearing doubt.
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +   [#permalink] 01 May 2018, 18:00
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