Exploring the Fixed Points of a Recursive Sequence and Their Behavior

In mathematics, the study of fixed points in recursive sequences can offer profound insights into the behavior of iterative processes. We will delve into the exploration of two fixed points of the given sequence defined by the equation x 98/x. This exploration will involve understanding the roots of the quadratic equation and observing the behavior of the sequence based on different initial seeds.

Initial Setup and Equation Analysis

Consider the equation:

x 98/x

To solve for x, we can rewrite the equation as:

x^2 98

Now, to find the fixed points of the sequence, we need to solve the quadratic equation:

x^2 - 9x - 8 0

The solutions to this quadratic equation are given by the quadratic formula:

x frac{9 pm sqrt{81 32}}{2} x frac{9 pm sqrt{113}}{2}

Thus, we obtain the two solutions:

x 9sqrt{113} / 2

and

x 9 - sqrt{113} / 2

Behavior of the Fixed Points

The quadratic equation x^2 - 9x - 8 0 has two roots, and these roots can be seen as the fixed points of the sequence defined by the equation:

a_n 98 / a_{n-1}

For a fixed point, it must satisfy the equation:

a_{n 1} a_n 98 / a_n

Which simplifies to:

a^2 - 9a - 8 0

Evaluating these fixed points can provide insight into how the sequence behaves for different initial seeds. Let's use Desmos to visualize the two possible fixed points:

Graph of the two fixed points

From the graph, we can see that the two solutions are approximately 9.815 and -0.815. However, these illustrations are two approximate solutions and require further analysis.

Convergence Analysis with Different Seeds

When examining the behavior of the sequence with different initial seeds, it is important to consider the convergence properties around these fixed points. The choice of which fixed point is preferred over the other often depends on the seed value.

To understand the convergence, let's define the sequence as:

a_1 x a_n 98 / a_{n-1}

For a fixed point, it must satisfy:

a_2 a_1 98 / a_1

Which leads to:

x^2 - 9x - 8 0

The fixed points are:

x 9 sqrt{113} / 2

and

x 9 - sqrt{113} / 2

Most people would choose the positive root, as it is more intuitive. However, if the seed is negative, the sequence can remain negative, leading to a constant sequence. Hence, the negative root is a valid fixed point but it is not the preferred choice.

Let's examine the use of these fixed points in a graph:

Graph of two functions yx and y98/x. The x-coordinate of the intersection point is the fixed point.

The fixed point as the seed results in a constant sequence. For other seeds, particularly close to the negative root, the sequence eventually converges to the positive root. This is due to the slope around the roots.

Convergence Based on Slope Analysis

Considering the calculus behind the convergence phenomenon, we need to analyze the slope of the curve around the fixed points. For the positive root:

f(9 sqrt{113}/2) approx -0.08

And for the negative root:

f(9 - sqrt{113}/2) approx -12

A small deviation around the negative root would cause the sequence to deviate and eventually converge to the positive root.

Conclusion

In summary, the fixed points of the sequence defined by x 98/x are x 9 sqrt{113} / 2 and x 9 - sqrt{113} / 2. The positive root is generally preferred due to its more intuitive and stable behavior when the initial seed is positive. The behavior of the sequence based on different seeds highlights the importance of convergence analysis in understanding recursive sequences.