Problem 1

For each integer

a_0>1

, define the sequence

a_0,a_1,a_2,\ldots

for

n\ge 0

a_{n+1}=\begin{cases}\sqrt{a_n} & \text{if }\sqrt{a_n}\text{ is an integer},\\ a_n+3 & \text{otherwise}.\end{cases}

Determine all values of

a_0

for which there exists a number

A

such that

a_n=A

for infinitely many values of

n

Step 5 of 5: Conclusion

Previous step Next step

a_0=3k,\ k\in\mathbb{Z}_{>0}

Detailed analysis

Combining the previous two steps with the equivalence from Step 1: a number $A$ occurs infinitely often exactly when $a_0$ is a positive multiple of $3$ , and then one may take $A=3$ , $A=6$ , or $A=9$ .