SOLUTION: Solve the equation for the value of n: {{{x^(n-1)+x^(n+1)=320}}} My approach was {{{x^(n-1)+x^(n+1)=320}}} {{{x^(n-1)+x^(n+1)=256+64}}} {{{x^(n-1)+x^(n+1)=2^6+2^8}}} So

Let start discussing the problem's formulation.


If you allow x and "n" to be any real numbers, then, probably, other solutions (non-integer, rational or irrational) may exist.


But, probably, the problem restricts you by integer values of x and "n" --- it would be very natural.


So, let's assume that we are looking for integer positive x and "n", only.


Then your approach is only PARTLY good : you just found one solution, but you did not guarantee that there is no other solution.

In your post, you even did not touch this issue.


In this sense, your solution is not complete.


There is another approach.


The starting equation can be transformed into other, equivalent equation


     = 320

     = .


Having it in this form, it is easy to prove that the only solution is


    x = 2, n = 7,


using the divisibility properties of integer numbers.