Suppose x=-1 and k is a positive number. If n is a positive root of t

Question

Suppose  x=-1  and  k  is a positive number. If  n  is a positive root of the equation  n^4 – 2^{4k} = 0 , what is the value of  x+x^n+x^{n+1}+x^{n+2} ?

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

chetan2u · Answer

FIRST step - solve for 'n'...
 n^4 – 2^{4k} = 0..........(n^2-2^{2k})(n^2+2^{2k})=0...............(n-2^{k})(n+2^k)(n^2+2^{2k})=0 
so what all can be the value of n...  2^k  or  -(2^k)  or  \sqrt{-(2^{2k})} , which is not possible
positive root of n is  2^k 
and since k is positive integer,  2^k  or n will also be EVEN..

x+x^n+x^{n+1}+x^{n+2} = (-1)+(-1)^{even}+(-1)^{even+1}+(-1)^{even+2}=(-1)+1+(-1)+1=0 
C

niks18 · Answer

n^4-2^{4k}=0=>n^4=2^{4k} . Taking fourth root of both sides we have

n=2^k=Even  as  k  is a positive integer(Note as  n  is a positive root, so  n=-2^k  is not possible here)

Hence  x+x^n+x^{n+1}+x^{n+2}=-1+(-1)^{even}+(-1)^{odd}+(-1)^{even}=0

Option  C

MathRevolution · Answer

=>

n^4 – 2^{4k} = 0 
 ⇔ (n^2-2^{2k})(n^2+2^{2k}) = 0 
 ⇔ (n^2-2^{2k}) = 0, since n^2+2^{2k} ≠ 0 
 ⇔ (n+2^k)(n-2^k) = 0 
 ⇔ n = 2^k  
Since k > 0, n  must be even.

Thus  n = 2m  for some integer  m . It follows that 
 x+x^n+x^{n+1}+x^{n+2} 
 = (-1)+(-1)^n+(-1)^{n+1}+(-1)^{n+2} 
 = (-1)+(-1)^{2m}+(-1)^{2m} +1+(-1)^{2m+2} 
 = (-1) + 1 + (-1) + 1 = 0

Therefore, the answer is C.
Answer: C

hero_with_1000_faces · Answer

I do not understand why do we need this complex calculation, when we know it for a fact that any No. multiplied by 2 is always an Even number and as "K" is positive, so 2^4k has to be an even no. and thus n^4 is even too. and that makes the process much easier.

niks18 · Answer

Hi  MathRevolution

shouldn't it be better to mentioned "k" as positive integer rather than positive number because positive number can be a decimal value which will create ambiguity.

we all have solved the question assuming "k" to be a positive integer

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Suppose x=-1 and k is a positive number. If n is a positive root of t

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Suppose x=-1 and k is a positive number. If n is a positive root of t

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