Mathematics

Problem

Let

x

y

and

z

be positive real numbers satisfying

\log_e y - \log_e x = \log_e z - \log_e y = k

for some constant

k

. (i) Show that

x

y

and

z

are in geometric progression and determine their common ratio in terms of

k

. (ii) If

x = 5

and

k = \ln 3

, find the value of

z

. (iii) Hence calculate the sixth term of this geometric progression.