What is the value of \(2 + \sqrt {2 + \sqrt {2 + \sqrt { \ldots \ldots \ldots } } } ?\)
4
Finding the Value of an Infinite Nested Radical Expression
The problem asks for the value of the infinite nested square root expression \(2 + \sqrt {2 + \sqrt {2 + \sqrt { \ldots \ldots \ldots } } }\).
To find the value of such expressions, we can set the entire expression equal to a variable, say \(x\), and then use the self-repeating nature of the expression to form an equation.
Let the given expression be equal to \(x\):
\( x = 2 + \sqrt {2 + \sqrt {2 + \sqrt { \ldots \ldots \ldots } } } \)
Notice that the part under the first square root, \( \sqrt {2 + \sqrt {2 + \sqrt { \ldots \ldots \ldots } } } \), is the same as the original expression \(x\), but without the initial '2'. So, we can say that \( \sqrt {2 + \sqrt {2 + \sqrt { \ldots \ldots \ldots } } } = x - 2 \). However, a more common approach is to observe that the expression under the first square root, \( \sqrt {2 + \sqrt {2 + \sqrt { \ldots \ldots \ldots } } } \), is equal to the original expression minus the initial constant '2'. Let's rephrase this carefully.
Let \( x = 2 + y \), where \( y = \sqrt {2 + \sqrt {2 + \sqrt { \ldots \ldots \ldots } } } \). Since the expression under the first square root is the infinite sequence itself starting again with 2, we can say that \( y = \sqrt{2 + y} \).
Substitute \( y = x - 2 \) back into the equation for \( y \):
\( x - 2 = \sqrt{2 + (x - 2)} \)
Simplify the expression under the square root:
\( x - 2 = \sqrt{x} \)
Now, to solve for \(x\), we need to eliminate the square root. We can do this by squaring both sides of the equation:
\( (x - 2)^2 = (\sqrt{x})^2 \)
Expand the left side and simplify the right side:
\( x^2 - 4x + 4 = x \)
Rearrange the terms to form a quadratic equation by moving all terms to one side:
\( x^2 - 4x - x + 4 = 0 \)
\( x^2 - 5x + 4 = 0 \)
Now we need to solve this quadratic equation. We can factor it:
\( (x - 1)(x - 4) = 0 \)
This gives two possible solutions for \(x\):
- \( x - 1 = 0 \implies x = 1 \)
- \( x - 4 = 0 \implies x = 4 \)
We need to check which of these solutions is valid in the context of the original expression. The original expression is \( x = 2 + \sqrt {2 + \sqrt {2 + \sqrt { \ldots } } } \). Since the square root of a non-negative number is always non-negative, the term \( \sqrt {2 + \sqrt {2 + \sqrt { \ldots } } } \) must be greater than or equal to 0. Therefore, \( x = 2 + (\text{non-negative value}) \) must be greater than or equal to 2.
Let's examine our solutions:
- If \( x = 1 \), this value is less than 2. So, \(x=1\) is not a valid solution for the original expression.
- If \( x = 4 \), this value is greater than or equal to 2. This is a plausible solution.
We can also verify the solutions in the equation we derived by squaring, \( x - 2 = \sqrt{x} \). Squaring can sometimes introduce extraneous solutions.
- For \( x = 1 \): Left side = \( 1 - 2 = -1 \). Right side = \( \sqrt{1} = 1 \). \( -1 \neq 1 \). So \( x=1 \) is not a valid solution.
- For \( x = 4 \): Left side = \( 4 - 2 = 2 \). Right side = \( \sqrt{4} = 2 \). \( 2 = 2 \). So \( x=4 \) is a valid solution.
Thus, the value of the expression \(2 + \sqrt {2 + \sqrt {2 + \sqrt { \ldots \ldots \ldots } } }\) is 4.
The value of the expression is 4.
Revision Table: Solving Infinite Nested Radicals
| Step | Description | Application to the Problem |
|---|---|---|
| Step 1 | Set the entire infinite expression equal to a variable (e.g., \(x\)). | \( x = 2 + \sqrt {2 + \sqrt {2 + \sqrt { \ldots } } } \) |
| Step 2 | Recognize that the repeating part of the expression can also be represented in terms of the variable. | \( x - 2 = \sqrt {2 + \sqrt {2 + \sqrt { \ldots } } } \). Let \(y\) be the nested part, \(y = \sqrt{2+y}\), and \(x = 2+y\). Then \(y = x-2\). So \(x-2 = \sqrt{2+(x-2)}\), which simplifies to \(x-2 = \sqrt{x}\). |
| Step 3 | Form an equation using this relationship. | \( x - 2 = \sqrt{x} \) |
| Step 4 | Solve the equation for the variable, usually by squaring both sides. | \((x-2)^2 = (\sqrt{x})^2 \implies x^2 - 4x + 4 = x \implies x^2 - 5x + 4 = 0\) |
| Step 5 | Solve the resulting equation (often a quadratic equation). | \((x-1)(x-4) = 0 \implies x=1\) or \(x=4\) |
| Step 6 | Check the validity of the solutions in the original expression or in the equation before squaring to discard extraneous solutions. | Since \(x = 2 + \sqrt{\dots}\), \(x\) must be \(\ge 2\). \(x=1\) is invalid. \(x=4\) is valid. Checking in \(x-2 = \sqrt{x}\), \(1-2 = -1 \neq \sqrt{1}=1\); \(4-2=2 = \sqrt{4}=2\). Valid solution is \(x=4\). |
Additional Information: Infinite Series and Radicals
Infinite nested radicals, like the one in this problem, are a type of infinite expression. The method used to solve them relies on the fact that the expression repeats itself indefinitely. This allows us to create an algebraic equation that can be solved.
Not all infinite nested radicals converge to a finite value. For an expression of the form \( \sqrt{a + \sqrt{a + \sqrt{a + \ldots}}} \), it converges if \(a \ge 0\). The value \(x\) satisfies \(x = \sqrt{a+x}\), leading to \(x^2 - x - a = 0\). The positive solution is \( x = \frac{1 + \sqrt{1+4a}}{2} \).
In our specific problem, the expression is \(2 + \sqrt {2 + \sqrt {2 + \sqrt { \ldots } } }\). This is slightly different as there's an initial '2' outside the first radical. If the expression was just \( \sqrt {2 + \sqrt {2 + \sqrt { \ldots } } } \), we would solve \( x = \sqrt{2+x} \implies x^2 - x - 2 = 0 \implies (x-2)(x+1) = 0 \). The positive solution is \(x=2\). However, our problem has an initial '2' added.
Let \( y = \sqrt {2 + \sqrt {2 + \sqrt { \ldots } } } \). From the above, \(y=2\). The original expression is \( 2 + y \). So, the value is \( 2 + 2 = 4 \).
This confirms the result obtained by setting the whole expression equal to \(x\).
Similar Questions
What is the square root of 16 + 6√7?
Which one of the following is an irrational number?
Consider the following statements in respect of two integers p and q (both > 1) which are relatively prime:
1. Both p and q may be prime numbers.
2. Both p and q may be composite numbers.
3. One of p and q may be prime and the other composite.
Which of the above statements are correct?For any two real numbers a and b, \(\sqrt {{{\left( {a - b} \right)}^2}} + \sqrt {{{\left( {b - a} \right)}^2}} \) is
If \(x = \frac{{1\; + \;\sqrt 3 }}{2}\) and y = x 3, then y satisfies which one of the following equations?
If \(a\; = \;\sqrt {7\; + \;4\sqrt 3 ,}\) then what is the value of a + 1/a?
What is \(0.\overline {53} + 0.5\overline {3} \) equal to?
If the points P and Q represent the real numbers 0.83 ̅ and 0.62 ̅ on the number line, then the distance between P and Q is
Which one of the following rational numbers has non-terminating and repeating decimal expression?
Important Questions from Rational or Irrational Numbers
Which of the following is the smallest fraction?
4/5, 7/8, 6/7, 5/6
When 0.232323.... is converted into a fraction, then it is equal to:
Which of the following number is irrational?
Which of the following numbers will have an irrational square root?
What is the square root of 16 + 6√7?
