Math 213 - Homework 8
Assigned: 10/10/07
Due: 10/17/07 at the start of class.
Notation: Exercise a.b.c(d) stands for part (d) of Exercise c from Section a.b.
Problems:
(1) 7.1.2(e).
(2) 7.1.14. [Part (c) is similar to solving the recurrence from example 5].
(3) 7.1.24(a).
(4) 7.1.28(a).
(5) Define w0= 1, and for n≥1, wn=w1·w2· · · wn−1+ 1. (That is, wnis one more than
the product of the previous terms). Show that wn=w2
n−1−wn−1+ 1 for n≥1.
(6) 7.2.6.
(7) 7.2.8.
(8) 7.2.18.
(9) 7.2.22.
(10) A frog jumps between two lillypads. It starts out on lillypad 1 at midnight. If the
frog is on lillypad 1 nminutes after midnight, there is a 50 percent chance it jump to
lillypad 2 and stay there for a minute, and a 50 percent chance it will stay on lillypad
1 for a minute. If the frog is on lillypad 2 nminutes after midnight, it will jump back
to lillypad 1 for a minute. Let anbe the probability that the frog is on lillypad 1 at the
end of nminutes (so a0= 1 and a1= 1/2). Show that for n≥2,
an= 1 −1
2an−1.
1