Infinite Nested Radical Equations
In this lesson, we explore an interesting type of equation known as an infinite nested radical equation. This kind of expression contains a repeating square root pattern that continues endlessly. Our goal is to find the value of x that satisfies the equation.
1. Theory of Infinite Nested Radicals
An infinite nested radical is an expression where the same square root structure repeats forever. Because the pattern never ends, we can take advantage of its self-similar nature by introducing a variable to represent the entire expression.
2. Concept Explanation
The key idea is that the expression inside the square root is identical to the whole expression itself. This allows us to rewrite the equation using substitution, which simplifies the problem into a solvable algebraic equation.
3. Steps or Formula
- Define a variable to represent the infinite nested expression.
- Use the repeating pattern to create a new equation.
- Substitute the known value into the equation.
- Square both sides to eliminate the square root.
- Solve the resulting linear equation.
4. Example Problem
Given the equation:
$$ \sqrt{2x - \sqrt{2x - \sqrt{2x - \ldots}}} = 20 $$Step 1: Define the nested expression.
Let the entire expression be represented by y.
$$ y = \sqrt{2x - \sqrt{2x - \sqrt{2x - \ldots}}} $$Since the value of the expression is given, we know that:
$$ y = 20 $$Step 2: Use the repeating pattern.
Because the pattern repeats, the expression inside the square root is also y.
$$ y = \sqrt{2x - y} $$Step 3: Substitute the known value.
$$ 20 = \sqrt{2x - 20} $$Step 4: Eliminate the square root.
$$ 20^2 = 2x - 20 $$ $$ 400 = 2x - 20 $$Step 5: Solve for x.
$$ 2x = 420 $$ $$ x = 210 $$5. Final Answer
The value of x that satisfies the infinite nested radical equation is 210.
This substitution technique is especially powerful when dealing with equations that contain infinite repeating structures, such as recursive formulas and feedback-based systems.
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