Infinite Nested Radical Equations
In this lesson, we study an interesting type of equation called an infinite nested radical equation. This equation contains a square root that repeats endlessly, forming a continuous pattern. Our goal is to determine 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 appears again and again. Because the pattern is infinite, we can simplify the problem by representing the entire expression with a single variable.
2. Concept Explanation
The main idea is recognizing that the expression inside the square root is identical to the whole expression. This self-similarity allows us to use substitution, turning a complex-looking equation into a simpler algebraic one.
3. Steps or Formula
- Let a variable represent the entire infinite radical.
- Rewrite the equation using the repeating pattern.
- Substitute the known value into the equation.
- Square both sides to remove the square root.
- Solve the resulting equation for x.
4. Example Problem
Given the equation:
$$ \sqrt{2x + \sqrt{2x + \sqrt{2x + \ldots}}} = 20 $$Step 1: Define the nested expression.
Let the entire infinite radical be represented by y.
$$ y = \sqrt{2x + \sqrt{2x + \sqrt{2x + \ldots}}} $$From the problem, we know that:
$$ y = 20 $$Step 2: Use the repeating structure.
Because the pattern repeats infinitely, 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.
$$ 2x = 380 $$ $$ x = 190 $$5. Final Answer
The value of x that satisfies the infinite nested radical equation is 190.
This substitution approach is very effective for solving equations with infinite repeating patterns, and it is commonly used in advanced mathematics involving recursive and self-referential expressions.
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