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

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