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If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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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
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Brent Hanneson – GMATPrepNow.com Sign up for our free Question of the Day emails
CEO
Joined: 12 Sep 2015
Posts: 2865
Location: Canada
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =\)
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
Answer:
Cheers,
Brent
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If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =\)
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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Posts: 6796
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =\)
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 + . . . +
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GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =\)
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
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If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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GMATPrepNow wrote:
If \(J = 2 + 4 + 6 + 8 + . . . 98 + 100\), and \(K = 1 + 3 + 5 + 7 + . . . + 97 + 99\), then \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =\)
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 + . . . +
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GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =\)
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
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Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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Answer is C .
We can pair any 2 consecutive term and apply (a+b) (a-b) in stead of a^2 - b ^2.
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Joined: 05 Mar 2018
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Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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GMATPrepNow wrote:
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =\)
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
Answer:
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...?
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Joined: 12 Sep 2015
Posts: 2865
Location: Canada
Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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MayurAgrawal wrote:
GMATPrepNow wrote:
GMATPrepNow wrote:
If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . + 97 + 99, then \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + . . . . . + 97^2 - 98^2 + 99^2 - 100^2 =\)
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
Answer:
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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Re: If J = 2 + 4 + 6 + 8 + . . . 98 + 100, and K = 1 + 3 + 5 + 7 + . . . +
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Quote:
Quote:
Hi Brent, I did't understand the step 3. Can you please elaborate, how did you write
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 + . . . + &nbs
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01 May 2018, 18:00
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42% (02:04) correct 
