Suppose another hiker takes 92 minutes to complete the hike .find the mean and media times including this new time

This is a geometric sequence with first term 1 and common ratio -1/2.  r=-1/2.

a(n) = a(1)*(r)^(n-1). 

Check:  If n=2 our formula must return -1/2.  Does it?

a(2) =    1(-1/2)^(2-1)  =  (-1/2)^1 = - 1/2.  Yes.
a(3) should be 1/4.  Is it?     a(3) = (-1/2)^(3-1) = 1/4    Yes. 

Then a(8) = (-1/2)^(8-1) = (-1/2)^7 = -1 / 2^7  = -1/128 (answer)