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Math Revolution and GMAT Club Contest! If x=-1 and n is the sum of the

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Math Revolution and GMAT Club Contest Starts!



QUESTION #13:

If x=-1 and n is the sum of the second 101 prime numbers, what is the value of \(x^n + x^{(n+1)} + x^{(n+2)} + x^{(n+3)}\)?

A. -2
B. -1
C. 0
D. 1
E. 2


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[Reveal] Spoiler:

Math Revolution and GMAT Club Contest

The Contest Starts November 28th in Quant Forum


We are happy to announce a Math Revolution and GMAT Club Contest

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Re: Math Revolution and GMAT Club Contest! If x=-1 and n is the sum of the [#permalink]

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New post 19 Dec 2015, 08:59
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There are total 25 prime numbers from 1 to 100
There are total 26 prime numbers from 1 to 101

Now given , " second 101 prime numbers " , I think that prime number starts from 3 to 101 . Excluding 2 [first prime number]

List of second 101 prime numbers are
3,5,7,11,13,17.....101
There are total 25 second 101 primer numbers.

If we can calculate n as even or odd, that will results into the solution as X = -1

Sum of second 2 prime numbers is even
Sum of second 3 prime numbers is odd
Sum of second 4 prime numbers is even
Sum of second 5 prime numbers is odd
.
.
.
.
Sum of second 25 prime numbers is odd

There n is odd.

x^n + x^(n+1) + x^(n+2) + x^(n+3)
is

(-1)^odd + (-1)^(odd + 1) + (-1)^(odd + 2) + (-1)^(odd + 3)

-1 + (-1)^(even) + (-1)^(odd) + (-1)^(even)

-1 + 1 - 1 + 1

0

Answer is C

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n=sum of second 101 prime numbers
Now , all prime numbers except 2 are odd. Therefore, sum of odd no of prime nos=odd and sum of even no of prime nos=even
Therefore, sum of second 101 primes nos(in which all are odd)=odd
or n=odd
Reqd equation= -1+1-1+1=0
Answer C

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New post 19 Dec 2015, 10:05
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Given: x=-1
n = Sum of the second 101 prime nos

In the second set of 101 prime nos,
1. There are no even nos
2. Sum of any two odd nos is a Even no

Since n is the sum of 101 prime nos, n = odd

Now,
(-1)^n+(-1)^(n+1)+(-1)^(n+2)+(-1)^(n+3)

As n = odd,
n+1= Even
n+2=Odd
n+3=Even
and (-1)^odd = (-1), and (-1)^Even = +1

On Solving,
-1+1-1+1=0

Alternatively, irrespective of whether n= odd/even, ans =0
Case 1: n = odd, n+1 = Even ,n+2 = odd, n+3 = Even (-1+1-1+1 = 0)
Case 2: n = Even, n+1 = odd, n+2 = Even, n+3 = Odd (-1+1-1+1 = 0)

Ans =C = 0

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First, the sum of second 101 prime numbers needs to be calculated.

Now, second 101 prime numbers means that the 101 prime numbers after 2. We also know that the sum of the 101 prime numbers after 2 will be an odd number since all these prime numbers are odd. => n is odd.

Now to find the value = (-1)^odd + (-1)^even + (-1)^odd + (-1)^even = -1 + 1 + (-1) + 1 = 0

Option C.
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If x=-1 and n is the sum of the second 101 prime numbers, what is the value of xn+x(n+1)+x(n+2)+x(n+3)?

A. -2
B. -1
C. 0
D. 1
E. 2

----

x = -1 ;
n = sum of second 101 "prime" numbers;

Given that all prime numbers above 2 are ODD , sum of 101 second prime numbers is a sum of ALL ODD INTEGERS,

odd integer + odd integer = even integer
Extending this observation, sum of EVEN NUMBER of odd INTEGERS is always EVEN,
sum of ODD number of ODD INTEGERS is always ODD.
Therefore, we need not really deduce the value of n, but we can infer that n is definitely an ODD integer.

Plugging it back into the equation, we're looking at

(-1)^(odd) + (-1)^(even) + (-1)^(odd) + (-1)^(even)
= -1 + (+1) + (-1) + (+1)
= 0

Correct answer : C
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Since n is the sum of 2nd 101 Prime Numbers, n Must be Odd

Therefore, (-1)^odd + (-1)^even + (-1)^odd + (-1)^even =-1 +1 -1 +1 = 0

Answer is C

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New post 21 Dec 2015, 00:45
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x = -1

n = Sum of second 101 prime numbers is odd. (odd + odd + .... 101 times = odd)

Now, the question can be rephrased as --> (-1)^odd + (-1)^even + (-1)^odd + (-1)^(even) = -1 + 1 - 1 + 1 = 0

Answer: C

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Answer is C.

xn+x(n+1)+x(n+2)+x(n+3) can be written as x^n(1+x+x^2+x^3). Apply x=-1 in this equation.

(-1)^n*(1-1+1-1) = 0. Hence answer is 0.
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Since n is an integer, in \(n, (n+1), (n+2), (n+3)\) there will be two odd numbers and two even numbers.
\(-1^{2k} = 1\) and \(-1^{(2k+1)} = -1\)
=> \(x^n + x^{(n+1)} + x^{(n+2)} + x^{(n+3)} = 0\), as it's the sum of two (-1) and two (1)
Answer C.
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We are looking for the sum of 4 consecutive powers of x.
- Since n in the sum of prime numbers, n is an integer. (in this case, excluding prime number 2, n is odd) however we don't need to know that.
- Since x is negative, the even powers will yield 1 and the odd powers -1.
Therefore the answer will always be zero: 1-1+1-1=0 and -1+1-1+1 =0
answer C
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The answer is 0
What ever may be the value of n ( either odd or even),
the expression x^n+x^n+1+x^n+2+x^n+3 will have two even power of x and two odd powers of x. Since x is -1, the expression reduces to +1+1-1-1=0 (order can vary )

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x=-1 n = the sum of the second 101 prime numbers, value of xn+x(n+1)+x(n+2)+x(n+3)
Except 2, all prime numbers are odd.So sum of all odd numbers 101 times is again odd . So n is odd number.

(-1)^odd number + (-1)^even number + (-1)^odd number + (-1)^even number = 0

which is answer choice C.

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New post 25 Dec 2015, 12:53
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Given: x = -1

x^n + x^(n+1) + x^(n+2) + x^(n+3) can be re-written as the following by taking x^n as common:

= x^n ( 1 + x + x^2 + x^3) - put the value of x

= (-1)^n * ( 1 + (-1) + (-1)^2 + (-1)^3)

= (-1)^n * 0

= 0

Option C is the correct answer.

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Re: Math Revolution and GMAT Club Contest! If x=-1 and n is the sum of the [#permalink]

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New post 27 Dec 2015, 11:21
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Bunuel wrote:

Math Revolution and GMAT Club Contest Starts!



QUESTION #13:

If x=-1 and n is the sum of the second 101 prime numbers, what is the value of \(x^n + x^{(n+1)} + x^{(n+2)} + x^{(n+3)}\)?

A. -2
B. -1
C. 0
D. 1
E. 2


Check conditions below:


[Reveal] Spoiler:

Math Revolution and GMAT Club Contest

The Contest Starts November 28th in Quant Forum


We are happy to announce a Math Revolution and GMAT Club Contest

For the following four (!) weekends we'll be publishing 4 FRESH math questions per weekend (2 on Saturday and 2 on Sunday).

To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific.
Then a week later, the forum moderator will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6-months access to GMAT Club Tests.

PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize:

PS + DS course with 502 videos that is worth $299!



All announcements and winnings are final and no whining :-) GMAT Club reserves the rights to modify the terms of this offer at any time.


NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process.

Thank you!



MATH REVOLUTION OFFICIAL SOLUTION:

The sum of the second 101 prime numbers is the sum of the all prime numbers from the second prime number to the next 101 prime numbers. Except 2, every prime number is an odd number. So, n=3+5+…………..+prime=odd.

Then, \(x^n + x^{(n+1)} + x^{(n+2)} + x^{(n+3)}=(-1)^{odd}+(-1)^{even}+(-1)^{odd}+(-1)^{even}=(-1)+1+(-1)+1=0\), and the correct answer is C.
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Re: Math Revolution and GMAT Club Contest! If x=-1 and n is the sum of the [#permalink]

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New post 07 Mar 2016, 03:00
[quote="Bunuel"][quote="Bunuel"]

Math Revolution and GMAT Club Contest Starts!



QUESTION #13:

If x=-1 and n is the sum of the second 101 prime numbers, what is the value of \(x^n + x^{(n+1)} + x^{(n+2)} + x^{(n+3)}\)?

A. -2
B. -1
C. 0
D. 1
E. 2

Answer is Zero Irrespective of the value of N
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New post 11 Jul 2016, 05:52
Probably a stupid question: What does "second" mean here? I got the question right but just by plugging in -1 due to gutfeeling :)

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Re: Math Revolution and GMAT Club Contest! If x=-1 and n is the sum of the [#permalink]

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New post 11 Jul 2016, 06:12
expertesp wrote:
Probably a stupid question: What does "second" mean here? I got the question right but just by plugging in -1 due to gutfeeling :)


Agree the wording could be clearer.

The sum of the second 101 prime numbers is the sum of the all prime numbers from the second prime number to the next 101 prime numbers. Except 2, every prime number is an odd number. So, n=3+5+…………..+prime=odd.
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New post 11 Jul 2016, 13:37
Thank you!

So add on to this what does "25 second 101 primer numbers" stand for?
Also one I figured out there is 25 prime numbers between 2 and 101 exclusive why do I plug in 1 for n? Which I did because of my gutfeeling but I rather have a mathematical explanation :)

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