If x = -1 and n is the sum of the second 101 prime numbers, what is th

Question

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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Official Answer

C

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dayasantoshi · Accepted Answer

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

Bunuel · Answer

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 .

riteshgmat · Answer

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

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

KS15 · Answer

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

abhib3388 · Answer

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 )

expertesp · Answer

Probably a stupid question: What does "second" mean here? I got the question right but just by plugging in -1 due to gutfeeling

Bunuel · Answer

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.

shaliny · Answer

hi Bunuel, here you mentioned that except 2, every prime no. is an odd, so n would be odd, how? it can be even because all are odd. I want to ask here should i check for no. of odd values between 3 to 101 to know n is odd or even?

Bunuel · Answer

Because you are adding 101 odd numbers.

All primes from the 2nd prime onward are odd, and you are summing exactly 101 of them. The sum of an odd count of odd numbers is odd, so n is odd. No need to count how many odds from 3 to 101.

shaliny · Answer

still confused about that i have to take prime no. up to 101 or total 101 prime numbers

Bunuel · Answer

It’s total 101 prime numbers (starting from the 2nd prime), not primes “up to 101.” So you are adding 101 odd primes, which makes the sum odd.

If x = -1 and n is the sum of the second 101 prime numbers, what is th

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