Question

If \[ x=\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}} \] then which one of the following is true?

• A. \(0<x<1\)
• B. \(1<x<2\)
• C. \(2<x<3\)
• D. \(3<x<4\)

Hint:

For repeating infinite radicals, replace the repeating part by the variable itself.

Answer 1

The Correct Option is C

Concept: For infinite nested radicals, let the whole expression be equal to itself.

Step 1: Assume the expression.
Let: \[ x=\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}} \] Since the pattern repeats: \[ x=\sqrt{5+x} \]

Step 2: Square both sides.
\[ x^2=5+x \] Rearrange: \[ x^2-x-5=0 \]

Step 3: Solve using quadratic formula.
\[ x=\frac{1\pm\sqrt{1+20}}{2} \] \[ =\frac{1\pm\sqrt{21}}{2} \] Since \(x>0\): \[ x=\frac{1+\sqrt{21}}{2} \]

Step 4: Approximate.
\[ \sqrt{21}\approx 4.58 \] So: \[ x=\frac{1+4.58}{2} \] \[ =\frac{5.58}{2} \] \[ \approx 2.79 \] Thus: \[ 2<x<3 \] Hence, the required answer is: \[ \boxed{2<x<3} \]