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, \(x_3 = 2x_2 −x_1 +1=2\times3−1+1=6\).
Find the values of \(x_4\) and \(x_5\).
- Find values of real constants \(A\), \(B\), \(C\) such that for \(n = 1\), \(2\), \(3\), \[\begin{equation*} x_n = A+Bn+Cn^2. \label{eq:star}\tag{$*$} \end{equation*}\]
Assuming that equation \(\eqref{eq:star}\) holds true for all \(n\geq1\), find the smallest \(n\) such that \(x_n\geq800\).
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 \(n\)?
