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Difficulty:
45%
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Question Stats:
70% (01:34) correct
30%
(01:26)
wrong
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If x = -1 and n is the sum of the first 100 prime numbers, what is the value of \(x^n + x^{n+1}+x^{n+2}\)?
A. -2
B. -1
C. 0
D. 1
E. 2
A. -2
B. -1
C. 0
D. 1
E. 2
24 Jan 2024, 07:20
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For this answer we don't need to know the first 100 prime number, neither the sum of it. We just need to know that 2, the only prime number that is an even number. Now the sum of an even and an odd number is an odd number.
So \(x^n\) where \(x\) is -1 & \(n\) is the sum of the first 100 prime numbers, which is an odd number, the answer will be -1. As -1^to any odd number is -1.
When \(x^n^+1\), \(n\) becomes an even number (odd + odd = even), the answer will be 1, as -1^any even number is 1.
Again \(x^n^+2\), the answer will be \(-1\).
So the sum of -1+1-1 = -1 (B)
So \(x^n\) where \(x\) is -1 & \(n\) is the sum of the first 100 prime numbers, which is an odd number, the answer will be -1. As -1^to any odd number is -1.
When \(x^n^+1\), \(n\) becomes an even number (odd + odd = even), the answer will be 1, as -1^any even number is 1.
Again \(x^n^+2\), the answer will be \(-1\).
So the sum of -1+1-1 = -1 (B)
24 Jan 2024, 07:47
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\(x^n+x^{n+1}+x^{n+2}\\
x^n(1+x+x^2)\\
x^n(1-1+1)\\
x^n\\
-1^n\)
Out of the first 100 prime numbers only "2" is the even prime number, rest 99 are odd prime number.
Sum of 1 Prime Even Number +99 Odd Prime Number = Odd Number
\(-1^{odd}\)
\(x^n+x^{n+1}+x^{n+2}\) = -1
x^n(1+x+x^2)\\
x^n(1-1+1)\\
x^n\\
-1^n\)
Out of the first 100 prime numbers only "2" is the even prime number, rest 99 are odd prime number.
Sum of 1 Prime Even Number +99 Odd Prime Number = Odd Number
\(-1^{odd}\)
\(x^n+x^{n+1}+x^{n+2}\) = -1
