If f (z) = m^z - n^z, and f (1) = 2, f (2) = 8. Find f (3).

If f (z) = m^z - n^z and f (1) = 2, f (2) = 8

then

m - n = 2 so m = n + 2

m^2 - n^2 = 8

substitute m = n + 2 into m^2 - n^2 = 8

so

m^2 - n^2 = 8

(n + 2) ^2 - n^2 = 8

n^2 + 4n + 4 - n^2 = 8

4n = 4

n = 1

m = n + 2 = 1 + 2 = 3

so f (z) = 3^z - 1^z

if z = 1 the f (1) = 3^1 - 1^1 = 2

if z = 2 the f (2) = 3^2 - 1^2 = 9 - 1 = 8

if z = 3 the f (3) = 3^3 - 1^2 = 27 - 1 = 26

answer

f (3) = 26