The IITJEE Mains are to be held on April 4 2015. So, let’s focus on some worthwhile basic problems.
Example. Let and let
for
. Compute
.
Solution:
At the first sight, you might think it’s a plain manipulation of recurring sequences. But, wait…let’s see…
With a little algebraic computation, we can show that this sequence has a period of 4; that is, for all
. But, why? We reveal the secret with trigonometric substitution; that is, we define
with
such that
. It is clear that if
is a real number, such an
is unique, because
is a bijection. Because
, we can rewrite the given condition as
, which is
by the addition and subtraction formulae. Consequently,
, or
(because tan has a period of
). In any case, it is not difficult to see that
for some integer k. Therefore,
, that is, the sequence
has a period 4, implying that
.
More later,
Nalin Pithwa