Solve the inequality `(-1/4)^n > -1/160000`

justaguide | Certified Educator

The inequality `(-1/4)^n > -1/160000` has to be solved.

`(-1/4)^n > -1/160000`

If n is even

=> `(1/4)^n > -1/160000`

This is true for all values of n as `(1/4)^n` is a positive number.

If n is odd

=> `-1*(1/4)^n > -1/160000`

=> `(1/4)^n < 1/160000`

`(log(160000))/(log 4) ~~ 8.6433`

=> `(1/4)^8.6433 ~~ 1/160000`

For all odd values of n greater than 8.6433 the inequality `(1/4)^n < 1/160000` holds.

The inequality is true for all even values of n and all odd values of n greater than 8.6433

aruv | Student

Question is not clear what about n is ? n is natural no. or integer or rational or irrationa number.

Let observe the question step by step.

If  n=0 ,then it is true because

`(-1/4)^0=1> -1/160000`

If n is natural number

(i) if n is even number then

`(-1/4)^n=1/4^n > -1/160000`

Also true.

(ii) if n=odd then

`(-1/4)^n=-1/4^n > -1/160000`

`1/160000> 1/4^n`

`1/625 >1/4^(n-4)`

Let n-4=m     ( m is natural number)

`4^m>625`

`=> m>=5`

i.e. n-4>=5

n>=9

If n is  negative integer

(i) if n= even =-l

then

`(-1/4)^(-l)=4^l> -1/160000`

It is true

(ii) if n=odd =-p

`(-1/4)^n=-4^p> -1/160000`

`4^p<1/160000`

`4^(p+4)<1/625`

which is not possible for any integral value of p .

For case of rational and irrational it is complecated.