Xn+1 = X + (1 - p 100 )Xn

Put X_n = Y_n - C then

Yn+1-C = X + (1 - p 100 )(Yn-C)

then choose C such that

-C = X - C(1 -

p 100

)

then

X =

-Cp 100

and

Yn+1 = (1 - p 100 )Yn

Hence

Yn+1 = (1 - p 100 )2Yn-1 = ... = (1 - p 100 )n+1Y0

and

Xn = (1 - p 100 )nY0 + 100X p .

Putting n=0, X₀=0 gives

Y0 = -

100X p

. Hence

Xn = 100X p (1 -(1- p 100 )n) ® 100X p

as n ® ¥ because

(1-

p 100

) < 1

.