Question 55
Let $\{x_{n}\}$ be a sequence defined by $x_{1}=2$ and
$$
x_{n+1}=x_{n}^{2}-x_{n}+1
$$
for $n \geq 1$. Prove that
$$
1-\frac{1}{2^{2^{n-1}}}\lt \frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{n}}\lt 1-\frac{1}{2^{2^{n}}}
$$
for all $n$.
$$
x_{n+1}=x_{n}^{2}-x_{n}+1
$$
for $n \geq 1$. Prove that
$$
1-\frac{1}{2^{2^{n-1}}}\lt \frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{n}}\lt 1-\frac{1}{2^{2^{n}}}
$$
for all $n$.