Finding integers and the limit of a sequence - from a 100 sequence and more (Russian book) Consider a sequence $a_{1}=1$ and for every $k>1$ integer $a_k=a_{k-1}+\dfrac{1}{a_{k-1}}$.
a) How many positive integers $n$ are there, satisfying that $a_n$ be an integer?
b) Find the limit (if there is) of the $a_k$ sequence!
Please help! It looks like that for a) the answer is 2. Thanks in advance!

At the limit point, $a_k=a_{k-1}$, hence $a_{k-1}=a_{k-1}+\frac{1}{a_{k-1}}$, which implies $\frac{1}{a_{k-1}}=0$, which can't exist, so $a_{k-1}\to \infty$, as does $a_k$.
Let $a_{k-1}=\frac{x}{y}$. Then: $a_k=\frac{x}{y}+\frac{y}{x}=\frac{x^2+y^2}{xy}$.
$$\frac{x^2+y^2}{xy}\in \Bbb Z \iff x=y, \text{where} \frac{x^2+x^2}{x^2}=2$$
Since $x\ne y$ for every successive case (we can infer this because it is an increasing function and tends to $\infty$), your only integer cases are $a_1=1$ and $a_2=2$.
