Understanding the dynamics of recursive sequences is crucial in both theoretical and applied mathematics. This article delves into a fascinating example involving a recurrence relation defined by:

1. Introduction to the Problem

Let's first establish the given recurrence relation:

$$X_{n 1} frac{sqrt{3} X_{n} - 1}{X_{n} sqrt{3}}$$

Starting with the initial condition $$X_1 3$$, we will investigate the value of $$X_{2011}$$.

2. Rearranging the Recurrence Relation

By rearranging the given recurrence relation, we can express it in a more insightful form:

$$X_{n 1} frac{X_n - frac{1}{sqrt{3}}}{1 - frac{1}{sqrt{3}}X_n} frac{X_n - tan{frac{pi}{6}}}{1 tan{frac{pi}{6}} cdot X_n}$$

Using the difference of angles identity for the tangent function, we obtain:

$$X_{n 1} tanleft(arctan X_n - frac{pi}{6}right)$$

This can be rewritten as:

$$arctan X_{n 1} - arctan X_n -frac{pi}{6} pi k text{ for some } k in mathbb{Z}$$

3. Summing the Recurrence Relation

By summing both sides over $$n 1, 2, ldots, 2010$$, we get a telescoping sum on the left side:

$$arctan X_{2011} - arctan X_1 2010 cdot left(-frac{pi}{6} pi kright) 2010k - 335pi$$

Given $$X_1 3$$, we have:

$$arctan X_{2011} arctan 3 2010k - 335pi$$

4. Applying the Tangent Function

Applying the tangent function to both sides and using the periodicity of the tangent function (with period $$pi$$), we can simplify the expression:

$$X_{2011} tan(arctan 3 2010k - 335pi)$$

Since $$tan(335pi theta) tan(theta)$$, the expression simplifies to:

$$X_{2011} tan(arctan 3) 3$$

Therefore, the value of $$X_{2011}$$ is 3.

5. Verification using MS Excel

Via Microsoft Excel, when we implement the given recurrence relation with the initial condition, we confirm that:

$$X_{2011} 3$$

This outcome supports our analytical solution, confirming the recursive nature of the sequence.

6. Conclusion

The exploration of the given recursive sequence demonstrates how the principle of telescoping sums and the properties of the tangent function can be used to derive elegant solutions. This problem showcases the beauty and complexity of recurrence relations in mathematical sequences.

Key Takeaways:

Recurrence Relation: A mathematical technique for defining sequences where each term is defined based on the previous one. Tangent Function: A trigonometric function used to solve problems involving angles and periodicity. Telescoping Sums: A simplification technique for sums where most intermediate terms cancel out.

Keywords: recursion, tangent function, telescoping sum