# Thread: sequence, not easy but not hard

1. ## sequence, not easy but not hard

A prime number $\displaystyle p$ is given. A sequence of positive integers $\displaystyle a_{1}, a_{2}, a_{3},...$ is determined by this conditon:
$\displaystyle a_{n+1}=a_{n}+p \lfloor \sqrt[p]{a_{n}} \rfloor$
Prove that there is a term in this sequence, which is a $\displaystyle p$-power of an integer number.

2. Anybody can help me ?? Is this so hard?

3. $\displaystyle a_1$ is a 1-power of an integer .

or have I misunderstood?

4. 1 is not prime number. It is harder question.

5. Originally Posted by Ununuquantium
A prime number $\displaystyle p$ is given. A sequence of positive integers $\displaystyle a_{1}, a_{2}, a_{3},...$ is determined by this conditon:
$\displaystyle a_{n+1}=a_{n}+p \lfloor \sqrt[p]{a_{n}} \rfloor$
Prove that there is a term in this sequence, which is a $\displaystyle p$-power of an integer number.
What do you mean by "p-power?" For example, choose p = 2 (to keep things simple.) Then given an $\displaystyle a_1 = 2$ (for instance) we get
$\displaystyle a_2 = a_1 + 2\sqrt{a_1} = 2 + 2\sqrt{2}$

$\displaystyle a_3 = a_2 + 2\sqrt{a_2} = 2 + \sqrt{2} + 2\sqrt{2 + \sqrt{2}}$

etc.

Are you saying that for some n of this sequence that $\displaystyle a_n = k^p$ where k is some integer?

-Dan

6. Yes exactly!!
$\displaystyle p$ power of an integer means
$\displaystyle a_{n}=k^{p}$ k integer, p prime