Note: My English is so bad but don’t worry about that! 🙂
A sequence is called linear of order two if there are real numbers
such that
In this topic we’ll use ideas in proofs of two following lemmas to solve some Olympiad problems.
lemma 1. If be sequence difined by
then
Proof. We need only prove .
For any , fixed
.
If then
and
.
If then
and
If then
therefore
and
.
Lemma 2. If a sequence of non-zero numbers satisfying for some number
:
then
Proof. From hypothesis we have (because they are equal to
), therefore
so
(*).
By (*) and we have
, so
(**).
Final, by (**) and we have
.
Now, before try to solve following problems, please read again two proves of two those lemmas.
Problem 1.
Let be sequence difined by
. Prove that
.
Problem 2.
Let be sequence difined by
. Find all positive integer numbers
such that
be perfect square.
Problem 3.
Let be sequence difined by
Prove that
.
Problem 4.
Let be sequence difined by
. Prove that
.
Problem 5.
Let be sequence difined by
a)Find the number of positive integer divisors of for any
.
b)Prove that for every the number
is a perfect square.
Problem 6.
The sequence of reals is difined by
Prove that
Problem 7.
be sequence difined by
. Prove that
Problem 8.
The sequence is defined by
. The sequence
is defined by
.
a)Prove that .
b)Show that the positive integers satisfy
iff
for some
.
Problem 9.
The sequence is defined by
. The sequence
is defined by
. Show that the non-negative integers
satisfy
iff
for some
.
Problem 10.
The sequence is defined by
. Prove that
.
Problem 11.
The sequence is defined by
. Prove that
is a perfect square for every
.
Problem 12.
The sequence of integers is defined by
. Prove that for every
,
is odd.
Problem 13.
Find the number of positive integer sequences such that
and
Problem 14.
Consider the sequence difined by
and
Prove that
.
Problem 15.
Let be positive reals. The sequence
is difined by
and
Prove that
iff
Problem 16.
Define . Prove that
is a perfect square for every positive integer
.
P.1 By lemma 2 , we have
.
, so we have
.
Alternatively ,
P.2
.
then
can’t be a perfect square .Follow the condition, easily to realize that
, and with
then
.
.
we get
.
By inductive we can prove that
Now , by some easy arguments, we have iff
So
Test for
done
The general problem.for
(*)
We have
Squared two sides of (*), we get :
Or:

.
, so
Similary:
So :
Alternatively,
Edit for P.2’solution :
by
.
On third line , move
Sorry ,Tuan.
to Quang: Not necessary use induction.
As lemma 1, we have
, or
therefore we have that identity!
Problem 17.
be a sequence with
,
and for all
,
Let
a) Prove that all the terms of the sequence are positive integers.
b) Prove that
is a perfect square for all positive integers
.