5. In this problem, as usual, let a = 5G, b = 90 a, where G is your academic group number.

a) Christian sells a box of his old toys to Jean, they are negotiating. Christian initially wanted to get C0 but Jean offered just J0. Here J0 < C0 are two positive real numbers. The negotiations go in rounds. Next time the boys meet each of them makes a step towards the other. Christian agrees to cross a% of the spead, i.e. the distance between their last round prices. Jean agrees to cross b% of the spread. For example, in the first round Christian's new sell price is C1 = C0 aS0/100, Jean's new buy price is J1 = J0 +bS0/100, where S0 = C0 J0. The n-th round prices are then given by

Cn =Cn1

aSn1 100

,

bSn1

Jn =Jn1 + , where Sn1 =Cn1 Jn1.

b) Show that the spread sequence {Sn} is a convergent geometric sequence.

c) Show that the sequences {Jn} and {Cn} are convergent and have the same limit L. Find the

expression for this limit L via the initial prices C0, J0.

d) Suppose that the negotiations stop when the spread value Sn is less than 0.01. How many

negotionation rounds will be required?