MAT 312/AMS 351, Summer 2011
Solutions to Homework Assignment 2
Maximal grade for HW2: 100 points
Section 1.2.
2. (15 points) Prove that for all positive integers
n
,
1 + 2
2
+ 3
2
+
...
+
n
2
=
n
(
n
+ 1)(2
n
+ 1)
6
.
(1)
Solution:
We prove this statement using induction by
n
.
Base.
We have to check the equation for
n
= 1:
1(1 + 1)(2
·
1 + 1)
6
=
2
·
3
6
= 1
.
Step.
Suppose that we proved the formula (1) for some number
n
, let us
prove it for
n
+ 1. We have:
1+2
2
+3
2
+
...
+
n
2
+(
n
+1)
2
= (1+2
2
+3
2
+
...
+
n
2
)+(
n
+1)
2
=
n
(
n
+ 1)(2
n
+ 1)
6
+(
n
+1)
2
=
n
+ 1
6
(
n
(2
n
+1)+6(
n
+1)) =
n
+ 1
6
(2
n
2
+7
n
+6) =
n
+ 1
6
(
n
+2)(2
n
+3) =
(
n
+ 1)((
n
+ 1) + 1)(2(
n
+ 1) + 1)
6
.
The last formula is precisely the right hand side of (1) with
n
replaced by
(
n
+ 1), so the equation (1) is true for (
n
+ 1) as well.
3. (15 points) The Fibonacci sequence is the sequence 1
,
1
,
2
,
3
,
5
,
8
,
13
...
where each term is the sum of the two preceding terms. Show that every two
successive terms of the Fibonacci sequence are relatively prime.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentSolution:
We can deﬁne the Fibonacci sequence as follows:
F
1
=
F
2
= 1
,
F
n
=
F
n

1
+
F
n

2
for
n >
2
.
Let us prove that
This is the end of the preview. Sign up
to
access the rest of the document.
 Summer '08
 BESCHER
 Algebra, Integers

Click to edit the document details