Question 131
A sequence of real numbers $x_{n}$ is an ordered list of reals in which the first number of the list is the term $x_{1}$, the second is the term $x_{2}$, and so on. For example, the usual sequence of positive integers can be described as $x_{n}=n$ for every positive integer $n$. Some sequences can be defined by recurrence equations, in which a term is defined as a function of the previous ones.
For example, the sequence of positive integers could be defined by $x_{1}=1$ and $x_{n}=x_{n-1}+1$ for every positive integer $n \geq 2$. In this way, we could compute $x_{2}=1+1=2$, $x_{3}=2+1=3$, and so on.
Consider a sequence of real numbers defined by $x_{1}=1$ and $x_{n}=x_{n-1}+\frac{1}{x_{n-1}^{2}}$ for every positive integer $n \geq 2$.
a) Compute $x_{2}, x_{3}$ and $x_{4}$.
b) Verify that the sequence is strictly increasing, that is, that $x_{n}\gt x_{n-1}$ for every positive integer $n$.
c) Notice that the sequence seems to grow very little. After computing a few initial terms, we might suspect that no term exceeds 2016, but in fact we will prove that there are terms greater than 2016. For this, we will use the auxiliary sequence $y_{n}=x_{n}^{3}$. Prove that $y_{n}\gt y_{n-1}+3$ for all $n \geq 2$.
d) Prove that there exists a number $N$ such that $x_{N}\gt 2016$.
For example, the sequence of positive integers could be defined by $x_{1}=1$ and $x_{n}=x_{n-1}+1$ for every positive integer $n \geq 2$. In this way, we could compute $x_{2}=1+1=2$, $x_{3}=2+1=3$, and so on.
Consider a sequence of real numbers defined by $x_{1}=1$ and $x_{n}=x_{n-1}+\frac{1}{x_{n-1}^{2}}$ for every positive integer $n \geq 2$.
a) Compute $x_{2}, x_{3}$ and $x_{4}$.
b) Verify that the sequence is strictly increasing, that is, that $x_{n}\gt x_{n-1}$ for every positive integer $n$.
c) Notice that the sequence seems to grow very little. After computing a few initial terms, we might suspect that no term exceeds 2016, but in fact we will prove that there are terms greater than 2016. For this, we will use the auxiliary sequence $y_{n}=x_{n}^{3}$. Prove that $y_{n}\gt y_{n-1}+3$ for all $n \geq 2$.
d) Prove that there exists a number $N$ such that $x_{N}\gt 2016$.