Year: 2008
Paper: 3
Question Number: 5
Course: LFM Pure
Section: Proof by induction
No solution available for this problem.
Most candidates attempted five, six or seven questions, and scored the majority of their total score on their best three or four. Those attempting seven or more tended not to do well, pursuing no single solution far enough to earn substantial marks.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1499.3
Banger Comparisons: 2
The functions ${\rm T}_n(x)$, for $n=0$, 1, 2, $\ldots\,$, satisfy
the recurrence relation
\[
{\rm T}_{n+1}(x) -2x {\rm T}_n(x) + {\rm T}_{n-1}(x) =0\,
\ \ \ \ \ \ \ (n\ge1).
\tag{$*$}
\]
Show by induction that
\[
\left({\rm T}_n(x)\right)^2 - {\rm T}_{n-1}(x) {\rm T}_{n+1}(x) = \f(x)\,,
\]
where $\f(x) = \left({\rm T}_1(x)\right)^2 - {\rm T}_0(x){\rm T}_2(x)\,$.
In the case $\f(x)\equiv 0$, determine (with proof) an expression
for ${\rm T}_n(x)$ in terms of ${\rm T}_0(x)$ (assumed to be non-zero)
and ${\rm r}(x)$, where
${\rm r}(x) = {\rm T}_1(x)/ {\rm T}_0(x)$.
Find the two possible expressions for ${\rm r}(x)$
in terms of $x$.
%Conjecture (without proof) the general form of the solution of $(*)$.
In terms of attempts and success, this resembled question 2. Apart from some that made no progress at all, the induction was accessible to many, as was the expression for Tₙ(x). In both of these there were frequent gaps or inaccuracies even though the solutions were understood in essence.