A sequence \(u_1, u_2, \ldots, u_n\) of positive real numbers is said to be unimodal if there is a value \(k\) such that
\[u_1 \leqslant u_2 \leqslant \ldots \leqslant u_k\]
and
\[u_k \geqslant u_{k+1} \geqslant \ldots \geqslant u_n.\]
So the sequences \(1, 2, 3, 2, 1\);\ \(1, 2, 3, 4, 5\);\ \(1, 1, 3, 3, 2\) and \(2, 2, 2, 2, 2\) are all unimodal, but \(1, 2, 1, 3, 1\) is not.
A sequence \(u_1, u_2, \ldots, u_n\) of positive real numbers is said to have property \(L\) if \(u_{r-1}u_{r+1} \leqslant u_r^2\) for all \(r\) with \(2 \leqslant r \leqslant n-1\).
- Show that, in any sequence of positive real numbers with property \(L\),
\[u_{r-1} \geqslant u_r \implies u_r \geqslant u_{r+1}.\]
Prove that any sequence of positive real numbers with property \(L\) is unimodal.
- A sequence \(u_1, u_2, \ldots, u_n\) of real numbers satisfies \(u_r = 2\alpha u_{r-1} - \alpha^2 u_{r-2}\) for \(3 \leqslant r \leqslant n\), where \(\alpha\) is a positive real constant. Prove that, for \(2 \leqslant r \leqslant n\),
\[u_r - \alpha u_{r-1} = \alpha^{r-2}(u_2 - \alpha u_1)\]
and, for \(2 \leqslant r \leqslant n-1\),
\[u_r^2 - u_{r-1}u_{r+1} = (u_r - \alpha u_{r-1})^2.\]
Hence show that the sequence consists of positive terms and is unimodal, provided \(u_2 > \alpha u_1 > 0\).
In the case \(u_1 = 1\) and \(u_2 = 2\), prove by induction that \(u_r = (2-r)\alpha^{r-1} + 2(r-1)\alpha^{r-2}\).
Let \(\alpha = 1 - \dfrac{1}{N}\), where \(N\) is an integer with \(2 \leqslant N \leqslant n\).
In the case \(u_1 = 1\) and \(u_2 = 2\), prove that \(u_r\) is largest when \(r = N\).