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Chapter 3: Problem 32

Consider \(x=\sqrt{5+x}\). (a) Apply the Fixed-Point Algorithm starting with \(x_{1}=0\) to find \(x_{2}, x_{3}, x_{4},\) and \(x_{5}\) (b) Algebraically solve for \(x\) in \(x=\sqrt{5+x}\). (c) Evaluate \(\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}}\)

Short Answer

Expert verified
(a) \(x_2 \approx 2.236\), \(x_3 \approx 2.689\), \(x_4 \approx 2.773\), \(x_5 \approx 2.788\). (b) \(x = \frac{1 + \sqrt{21}}{2}\). (c) \(\sqrt{5 + \sqrt{5 + \sqrt{5 + \cdots}}} = \frac{1 + \sqrt{21}}{2}\).

Step by step solution

01

Understand the Fixed-Point Algorithm

The Fixed-Point Algorithm is an iterative method where we repeatedly apply a function to an initial guess to get successive approximations that converge to a fixed point. In this case, the function is derived from the equation \(x = \sqrt{5+x}\), which can be rearranged to the form \(g(x) = \sqrt{5+x}\). We start with an initial guess \(x_1 = 0\) and find subsequent values \(x_2, x_3, x_4,\) and \(x_5\) using \(x_{n+1} = g(x_n)\).
02

Calculate x_2

Apply the function \(g(x) = \sqrt{5+x}\) starting with \(x_1 = 0\):\[x_2 = g(x_1) = \sqrt{5+0} = \sqrt{5} \approx 2.236\]
03

Calculate x_3

Use \(x_2\) to find \(x_3\):\[x_3 = g(x_2) = \sqrt{5 + 2.236} \approx \sqrt{7.236} \approx 2.689\]
04

Calculate x_4

Use \(x_3\) to find \(x_4\):\[x_4 = g(x_3) = \sqrt{5 + 2.689} \approx \sqrt{7.689} \approx 2.773\]
05

Calculate x_5

Use \(x_4\) to find \(x_5\):\[x_5 = g(x_4) = \sqrt{5 + 2.773} \approx \sqrt{7.773} \approx 2.788\]
06

Algebraically Solve x = sqrt(5 + x)

Square both sides of the equation: \(x^2 = 5 + x\). Rearrange to form a quadratic equation: \(x^2 - x - 5 = 0\). Solve using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -1\), \(c = -5\):\[x = \frac{1 \pm \sqrt{1 + 20}}{2} = \frac{1 \pm \sqrt{21}}{2}\]Since \(x\) represents a positive number, choose the positive root: \(x = \frac{1 + \sqrt{21}}{2}\).
07

Evaluate the Infinite Nested Radical

Recognize that the expression \(\sqrt{5 + \sqrt{5 + \sqrt{5 + \cdots}}}\) represents the fixed point of \(x = \sqrt{5 + x}\), which we found to be \(x = \frac{1 + \sqrt{21}}{2}\). Therefore, the infinite nested radical evaluates to this value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Iterative method
The iterative method is a mathematical approach where processes are repeated over and over to reach a goal. In simple terms, it means doing something again and again, getting closer to the best answer each time. In the context of solving equations, this method helps us find a 'fixed point' where repeated application of a function stops changing the value. This is especially useful when solving functions like the one given: \(x = \sqrt{5+x}\).

Here's how it works:
  • Start with an initial guess. In our exercise, that's \(x_1 = 0\).
  • Use the function derived from the equation, \(g(x) = \sqrt{5+x}\), to calculate the next approximation.
  • For example, the second guess (\(x_2\)) is \(\sqrt{5+0} = \sqrt{5}\), and the third guess (\(x_3\)) is \(\sqrt{5+\text{value of } x_2}\).
  • Keep applying the function to get new values: \(x_4, x_5\), etc.
Through these iterations, we converge on the fixed point of our function, which means that's where the value stops changing significantly.
Quadratic equation
Let's talk about quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). These types of equations are very common, and they describe a parabolic curve on a graph. In our exercise, we encounter a quadratic equation when we rearrange \(x = \sqrt{5+x}\).

Here's the breakdown:
  • First, we need to "undo" the square root by squaring both sides of \( x = \sqrt{5+x} \), resulting in: \( x^2 = 5 + x \).
  • This can be rearranged to the standard form \( x^2 - x - 5 = 0 \), which is a quadratic equation.
  • To find solutions to the quadratic equation, we use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \(a = 1\), \(b = -1\), \(c = -5\).
When applied, this formula provides two potential solutions. Since \(x\) represents a positive number in our original equation, we select the positive possible root for the solution.
Infinite nested radical
An infinite nested radical is an expression that goes on forever, and it's written like \(\sqrt{5 + \sqrt{5 + \sqrt{5 + \cdots}}} \). It may look complex at first, but it's simply a fancy form of a repeating pattern.

Some key points about infinite nested radicals include:
  • They usually converge to a fixed value. This value is equivalent to the solution of the equation from which the nested pattern starts.
  • In our example, if we set \(x\) to be this infinite nested radical, it satisfies the equation \( x = \sqrt{5 + x} \).
  • The solution to this, \( x = \frac{1 + \sqrt{21}}{2} \), is both the fixed point of the equation and the evaluation of the infinite nested radical.
Thus, even though the expression continues indefinitely, it has a specific value where it "settles down." Understanding this concept helps simplify seemingly infinite problems. Remember, the solution to these kinds of radicals comes from solving the corresponding fixed point equation!

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