k=1

In this problem letkbe the 3-rd non-zero digit of your student ID counting from the end of it, k=1. Let a= 10k >0 andb= 90a >0.

a)Alice and Bob are negotiating a price. Alice is a buyer, her initial offer isA₀. Bob is a seller, his starting sell price isB₀. HereA₀< B₀are two positive real numbers. The negotiations go in rounds. In the first round Alice agrees to crossa% of the distanceD₀=B₀A₀, i.e. her new offer isA₁=A₀+a*D₀/100. Bob, on the other hand, is only willing to crossb% of the difference, i.e. his new sell price isB₁=B₀b*D₀/100. In all the further rounds the story repeats: Alice is ready to crossa% of the distance between the last round prices, Bob is ready to crossb% of the distance:

D_n1=B_n1A_n1, A_n=A_n1+a*D_n1/100, B_n=B_n1b*D_n1/100.

b)Show that the difference sequenceD_nis a convergent geometric sequence.

c)Show that the sequences{A_n}and{B_n}are convergent. Moreover, they have the same limitL.