Review question

# How are these recursively defined sequences related? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6555

## Question

A list of real numbers $x_1$, $x_2$, $x_3$, … is defined by $x_1 = 1$, $x_2 = 3$ and then for $n\geq3$ by $x_n = 2x_{n−1} −x_{n−2} + 1.$

So, for example, $x_3 = 2x_2 −x_1 +1=2\times3−1+1=6$.

1. Find the values of $x_4$ and $x_5$.

2. Find values of real constants $A$, $B$, $C$ such that for $n = 1$, $2$, $3$, $\begin{equation*} x_n = A+Bn+Cn^2. \label{eq:star}\tag{*} \end{equation*}$
3. Assuming that equation $\eqref{eq:star}$ holds true for all $n\geq1$, find the smallest $n$ such that $x_n\geq800$.

4. A second list of real numbers $y_1$, $y_2$, $y_3$, … is defined by $y_1 = 1$ and $y_n = y_{n−1} + 2n.$

Find, explaining your reasoning, a formula for $y_n$ which holds for $n\geq2$.

What is the approximate value of $\dfrac{x_n}{y_n}$ for large values of $n$?