COT 3100H Spring 2008 Homework #6
Assigned: 2/21/08 (Thursday)
Due: 2/28/08 (Thursday)
1) Use induction on n to prove the following for all positive integers n:
n
 i2
i
 2  (n  1)2 n 1
i 1
2) Matrix multiplication for 2x2 matrices is defined as follows below:
f   ae  bg

h   ce  dg
 a b  e


 c d  g
af  bh 

cf  dh 
 2 5  1 4  12  7 

  
 .
For example, 
  1 3  2  3   5  13 
n
a b 
 simply indicates multiplying the given matrix n times.
Also, the notation 
c d 
Using induction on n, prove for all positive integers n that
n 
 2 1
n 1

  

 1 0
  n  n  1
n
3) Using induction on n, prove that 7 | (32n+1 + 2n+2) for all integers n  0.
4) Consider a sequence of numbers defined as follows:
t(0) = 1, t(1) = 4, t(n) = 5t(n-1)-6t(n-2), for all integers n > 1.
Using strong induction on n, prove for all non-negative integers n
that t(n) = -2n + 2(3)n.
d
dv
du
(uv)  u
v
) and the basic
dx
dx
dx
d 1
d n
( x )  1 , prove the power rule,
( x )  nx n 1 , for all positive
information that
dx
dx
integers x. (If you haven't had Calculus, you can skip this question.)
5) Assuming the product rule for derivatives (
Note: No programming option this time. It's important you learn induction =)