Equations
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Rahul@gurome
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I think something is missing in the equations. Can you please check that?
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Rahul@gurome
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(A) -2^n is always negative for all values of n, while (-2)^-n is positive for even values and negative for odd integers.selango wrote:Each of the following equations has at least one solution EXCEPT
A. -2^n = (-2)^-n
B. 2^-n = (-2)^n
C. 2^n = (-2)^-n
D. (-2)^n = -2^n
E.(-2)^-n = -2^-n
If n = 0, then -2^n = -1 and (-2)^0 = 1, so for n = 0, it does not hold true.
If n = 1, then -2^n = -2 and (-2)^-n = -1/2, so -2^n is not equal to (-2)^-n and if n = 2, then -2^n = -4 and (-2)^-n = 1/4, again-2^n is not equal to (-2)^-n. This equation gives no solution for any value of n.
(B) If n = 0, then 2^-n = 2^0 =1 and (-2)^n = (-2)^0 = 1. So for n = 0, it holds true.
(C) If n =0, 2^n == 2^0 = 1 and (-2)^-n = (-2)^0 = 1. So for n = 0, it holds true.
(D) If n = 0, (-2)^n = (-2)^0 = 1 and -2^n = -2^0 = -1. But for any odd value, say n = 1, (-2)^1 = -2 and -2^n = -2^1 = -2. So, (-2)^n = -2^n holds true for odd values.
(E) If n = 0, (-2)^-n = (-2)^0 = 1 and -2^-n = -2^0 = -1. If n = 1, (-2)^-1 = -1/2 and -2^-n = -2^-1 = -1/2. So, (-2)^n = -2^n holds true for odd values.
[spoiler]The correct answer is (A).[/spoiler]
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jainnikhil02
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IMO C
what is OA
what is OA
Nikhil K Jain
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goalevan
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A. -2^n = (-2)^-n
(-1) * 2^n * (-1)^n * 2^n = 1
(-1)^n * 2^(2n) = -1
Since 2^(2n) is always positive, n must be odd for (-1)^n * 2^(2n) to have a chance at being -1. For 2^(2n) = 1, n must be 0, an even number.
B. 2^-n = (-2)^n
(-1)^n * 2^n * 2^n = 1
(-1)^n * 2^(2n) = 1
Satisfied when n = 0.
C. 2^n = (-2)^-n
(-1)^n * 2^n * 2^n = 1
(-1)^n * 2^(2n) = 1
Satisfied when n = 0.
D. (-2)^n = -2^n
(-1)^n * 2^n = (-1) * 2^n
(-1)^n = -1
Satisfied whenever n is odd.
E. (-2)^-n = -2^-n
(-1) * 2^n = (-1)^n * 2^n
-1 = (-1)^n
Satisfied whenever n is odd.
Notice that the equations in answer choices B/C and D/E are algebraically the same. Being able to recognize this quickly would rule out all answers choices except for A.
(-1) * 2^n * (-1)^n * 2^n = 1
(-1)^n * 2^(2n) = -1
Since 2^(2n) is always positive, n must be odd for (-1)^n * 2^(2n) to have a chance at being -1. For 2^(2n) = 1, n must be 0, an even number.
B. 2^-n = (-2)^n
(-1)^n * 2^n * 2^n = 1
(-1)^n * 2^(2n) = 1
Satisfied when n = 0.
C. 2^n = (-2)^-n
(-1)^n * 2^n * 2^n = 1
(-1)^n * 2^(2n) = 1
Satisfied when n = 0.
D. (-2)^n = -2^n
(-1)^n * 2^n = (-1) * 2^n
(-1)^n = -1
Satisfied whenever n is odd.
E. (-2)^-n = -2^-n
(-1) * 2^n = (-1)^n * 2^n
-1 = (-1)^n
Satisfied whenever n is odd.
Notice that the equations in answer choices B/C and D/E are algebraically the same. Being able to recognize this quickly would rule out all answers choices except for A.
