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Each of the following equations has at least one solution EXCEPT

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Joined: 06 Sep 2013
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Re: Each of the following equations has at least one solution EXCEPT  [#permalink]

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New post 15 Jan 2014, 08:57
nverma wrote:
marcusaurelius wrote:
Each of the following equations has at least one solution EXCEPT

–2^n = (–2)^-n
2^-n = (–2)^n
2^n = (–2)^-n
(–2)^n = –2^n
(–2)^-n = –2^-n


IMHO A

a) –2^n = (–2)^-n
–2^n = 1/(–2)^n
–2^n * (–2)^n = 1, Keep it. Let's solve the other options..!!

b) 2^-n = (–2)^n
1/2^n = (–2)^n
1 = (–2)^n * (2^n)
For n=0, L.H.S = R.H.S

c) 2^n = (–2)^-n
2^n = 1/ (–2)^n
(2^n) * (–2)^n = 1
For n=0, L.H.S = R.H.S

d) (–2)^n = –2^n
(–2)^n / –2^n = 1
For n=1, L.H.S = R.H.S

e) (–2)^-n = –2^-n
1/ (–2)^n = 1/–2^n
For n=1, L.H.S = R.H.S


Why did you plug n=1 for the last two, wouldn't it be easier just to plug n=0 for all and see that A has no solution?
Just want to know if there was any specific reason why you did so

Thank you
Cheers
J :)

PS. Would be nice if we could get this question in code format!
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Re: Each of the following equations has at least one solution EXCEPT  [#permalink]

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New post 15 Jan 2014, 19:55
jlgdr wrote:
nverma wrote:
marcusaurelius wrote:
Each of the following equations has at least one solution EXCEPT

–2^n = (–2)^-n
2^-n = (–2)^n
2^n = (–2)^-n
(–2)^n = –2^n
(–2)^-n = –2^-n


IMHO A

a) –2^n = (–2)^-n
–2^n = 1/(–2)^n
–2^n * (–2)^n = 1, Keep it. Let's solve the other options..!!

b) 2^-n = (–2)^n
1/2^n = (–2)^n
1 = (–2)^n * (2^n)
For n=0, L.H.S = R.H.S

c) 2^n = (–2)^-n
2^n = 1/ (–2)^n
(2^n) * (–2)^n = 1
For n=0, L.H.S = R.H.S

d) (–2)^n = –2^n
(–2)^n / –2^n = 1
For n=1, L.H.S = R.H.S

e) (–2)^-n = –2^-n
1/ (–2)^n = 1/–2^n
For n=1, L.H.S = R.H.S


Why did you plug n=1 for the last two, wouldn't it be easier just to plug n=0 for all and see that A has no solution?
Just want to know if there was any specific reason why you did so

Thank you
Cheers
J :)

PS. Would be nice if we could get this question in code format!


We need to find the equation that has no solution. What we are trying to do is find at least one solution for 4 equations. The fifth one will obviously not have any solution and will be our answer.
Options (D) and (E) do not have 0 as a solution.
So you try n = 1 on (A), (D) and (E).
n = 1 is still not a solution for (A) but it is for (D) and (E).

(D) (–2)^n = –2^n
When you put n = 0, you get
(-2)^0 = -2^0
1 = -1 which doesn't hold.
So you try n = 1
(–2)^1 = -2^1
-2 = -2
n = 1 is a solution.

Same logic for (E)
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Re: Each of the following equations has at least one solution EXCEPT  [#permalink]

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New post 19 Feb 2018, 13:30
How do you determine that we should use 0 or 1? I used 2 and 3 and found all of them to not equate.
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Re: Each of the following equations has at least one solution EXCEPT  [#permalink]

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New post 19 Feb 2018, 21:08
hudhudaa wrote:
How do you determine that we should use 0 or 1? I used 2 and 3 and found all of them to not equate.


0, 1, and -1 are special cases. Usually we start with them in case we are looking at plugging numbers.
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Re: Each of the following equations has at least one solution EXCEPT &nbs [#permalink] 19 Feb 2018, 21:08

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