**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 .

Alternatively ,, so we have .

P.2

By inductive we can prove that .

Now , by some easy arguments, we have iff then can’t be a perfect square .Follow the condition, easily to realize that , and with then .

So .

Test for we get .

done

The general problem.for

We have (*)

Squared two sides of (*), we get :

Or:

Similary:

So :

.

Alternatively, , so

Edit for P.2’solution :

On third line , move by .

Sorry ,Tuan.

to Quang: Not necessary use induction.

As lemma 1, we have , or therefore we have that identity!

Problem 17.

Let be a sequence with , and for all ,

a) Prove that all the terms of the sequence are positive integers.

b) Prove that is a perfect square for all positive integers .