CS173 Discrete Mathematical Structures
Spring 2007
Homework #7
Due Sun 03/04/07, 8AM.
Solutions, Grading rubric (41 points)
1) (10 Points, For each problem: 1 point for base case, 1 point for the induction hypothesis, 3
points for the step. )
Let f
i
be the i-th Fibonacci number. Prove that:
1. f
21
+f
22
+... +f
2n
=f
n
f
n+1
.
2. f
0
f
1
+f
1
f
2
+f
2
f
3
+...+f
2n-1
f
2n
=f
22n
. (Hint: note that 2n is always even!)
Solutions:
1.1)
Base case: n=1.
f
21
= f
1
f
2
1
2
= 1*1
IH: for all k<=n, f
21
+f
22
+... +f
2n
=f
n
f
n+1
holds.
Step: nn+1
f
21
+f
22
+... +f
2n
+f
2n+1
=[IH]= f
n
f
n+1
+f
2n+1
= f
n+1
(f
n
+f
n+1
)= f
n+1
f
n+2
1.2)
Base case
n=0.
f
0
f
1
=f
20
1=1
n=1 (not really needed)
f
0
f
1
+f
1
f
2
=f
22
0*1+1*1=1*1
IH: for all k<=n, f
0
f
1
+f
1
f
2
+f
2
f
3
+...+f
2n-1
f
2n
=f
22n
holds.
Step: nn+1
(f
0
f
1
+f
1
f
2
+f
2
f
3
+...+f
2n-1
f
2n
)+f
2n
f
2n+1
+f
2n+1
f
2n+1
=[IH]= f
22n
+f
2n
f
2n+1
+f
2n+1
f
2n+2
= f
2n
(f
2n
+f
2n+1
)+f
2n+1
f
2n+2
=
f
2n
f
2n+2
+f
2n+1
f
2n+2
= f
2n+2
(f
2n
+f
2n+1
) = f
22n+2
2) (5 points. 1 point for base case, 1 point for the induction hypothesis, 3 points for the step.)
Use strong induction to prove that any non-prime positive integer number n can be written as a
product of primes.
Solution:
Base case: 4=2*2.
IH: Assume that any non-prime number x<n can be written as a product of primes.
We need to prove that if the IH holds and n is non-prime, n can be written as a product of primes.
Since n is non-prime, it can be written as a product of two numbers n=a*b, where a,b≠1,n.
Therefore, a,b<n. Since both a and b can be written as a product of primes (by IH), n can be written
too by just multiplying the prime factorizations of a and b.
