I want to know if this can be proved that it is true.

Question

I suppose we know that x > 0 . m is an integer and w is real number , where x = (1 + w)^m  ,0<=w<=0.5

Answer 1

Let me take a dig on this by assuming certain things. First I am going to reframe the question so that it can be in a "provable" format that you may be looking for.

 

If m is an integer and w is a real number 0<=w<=0.5, can we prove x>0 where x = (1+w)^m?

 

With that as the question, I am going to try and answer this.

 

w is a real number between 0 and 0.5. Let us take two cases where w=0 and then w>0 but <=0.5.

 

Case I: w=0

 

x =(1+0)^m = 1^m . For any integral value of m (three sub cases m=0, m<0 and m>0), x will be >0. To be precise, x will be equal to 1 for all integral values of m.

 

Case II: 0<w<=0.5. Here w is a positive integer greater than zero and less than or equal to 0.5.

 

(1+w)>1 since w is a positive real number greater than zero. To be precise, 1<(1+w)<=1.5

 

Hence,

 

x=(>1)^m . For any integral value of m (three sub cases m=0, m<0 and m>0), x will be >0 .

 

Hope this helps.