Math 312 Solutions To Practice Problems Worksheet With Answers
ADVERTISEMENT
1
2
3Solutions to Practice Problems, Math 312
1 Prove that the fourth power of an odd integer is expressible in the
form 16n + 1 for n
Z.
Solution If m is odd, we can write m = 2k + 1 for some k
Z and so
4
4
4
3
2
m
= (2k + 1)
= 16k
+ 32k
+ 24k
+ 8k + 1
and so we have
k(3k + 1)
4
3
m = 16 k
+ 2k
+
+ 1.
2
(3 +1)
It remains to show that
Z. If k is even, then
Z, while if k
2
2
3 +1
is odd, then 3k + 1 is even and so
Z. This completes the proof.
2
2 Define a by a
= 1, a
= 2, a
= 4 and
0
1
2
a
= a
+ a + a
, for n
1.
+2
+1
1
Show that a
2 for all n
N.
Solution Note the typo - I meant to write a
= 4. We use induction
2
on n. The inequality is true for n = 0, 1 and 2. Suppose that it is true
for all n
k where k
2. Then
1
2
2
+1
a
= a + a
+ a
2 + 2
+ 2
= 7 2
< 2
.
+1
1
2
3 If F is the nth Fibonacci number, prove that
2
F
F
F
= ( 1) .
+1
1
Solution We use Binet’s formula and find that the left-hand-side is
2
+1
+1
1
1
α
β
α
β
α
β
5
5
5
and hence is equal to
+1
1
1
+1
2(αβ)
α
β
α
β
.
5
Since αβ =
1, this is equal to
1
1
2
2
2
αβ
α
β
2 + α
+ β
( 1)
= ( 1)
= ( 1) .
5
5
4 Let a and n be positive integers with a > 1. Prove that, if a + 1
is prime, then a is even and n is a power of 2.
Solution If a + 1 is prime, then, since a > 1, it follows that a + 1
must be odd and so a must be even. If n > 1, then if n is not a power
1
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education