If p is an integer and m= -p + (-2)^p, is m^3 = -1

Question

If p is an integer and  m= -p + (-2)^p , is  m^3 \geq -1

(1) p is even

(2)  p^3 \leq -1

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chetan2u · Answer

m= -p + (-2)^p 
 If p is even, m will always be positive...
If p<0 and even, -p and (-2)^p both will be positive, so m is positive
If p<0 and odd, -p is positive and (-2)^(-p) will be negative but a fraction that will be less than -p

(1) p is even
Let p=0,  m= -p + (-2)^p=-0+(-2)^0=0+1=0 
Let p=-2,  m= -(-2) + (-2)^{-2}=2+\frac{1}{4} 
m>0...answer is YES

(2)  p^3 \leq -1 
means  p \leq -1 ..
 m= -p + (-2)^p ... both -p and  (-2)^p are positive if p is even.
 m= -p + (-2)^p =positive+positive=positive
If p is odd -p will be positive and greater than (-2)^-p

D

gvij2017 · Answer

Hello Sir!

I have a doubt for statement "If p<0, -p and (-2)^p both will be positive, so m is positive"
Take p = -3
(-2)^-3= -1/8 which is negative.

But the sum of -p+(-2)^p will always be positive as we move towards more negative numbers.

minustark · Answer

Let p=-2, m=−(−2)+(−2)−2=2+1/4

I'm a bit confused with this line. Shouldn't it be 2 - 1/4?

gvij2017 · Answer

Square or even power of any number or digit is always non-negative.

ShankSouljaBoi · Answer

worked on stem a bit,

m^3 >= -1
means -1<=m < infinity
or is -1<= -p+ (-2)^p < infinity ??????

A substitue -1/2 -5 and 3 , all satisfy hence correct.

B p^3 <= -1 means p<= -1 . Hence -p = positive + (-2)^ something negative = positive + 1/(-2)^positive .

Also -p will always be greater than 1/ (-2)^p
If p is even negative lets say -2 expression becomes positive + 1/(-2) positive = positive + positive .
Hence yes m >= -1

If p is odd negative lets say -1 expression becomes 1 -1 = 0 .
Hence yes m>= -1

D

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If p is an integer and m= -p + (-2)^p, is m^3 >= -1

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