Nested Radicals/Fractions

Look carefully at the expression. A nested radical such as $x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}$ has the property that the expression under the outermost radical is $2 + x$ itself — the tail of the infinite expression is identical to the whole. Similarly, the infinite continued fraction $x = 1 + \dfrac{1}{1 + \dfrac{1}{1 + \cdots}}$ satisfies $x = 1 + \dfrac{1}{x}$. Spotting this self-similarity is the crucial first step.

Let $x$ represent the value of the full nested expression. Replace the 'inner copy' with $x$ to obtain a clean algebraic equation. For the nested radical above: since the part under the root equals $2 + x$, write $x = \sqrt{2 + x}$. For the continued fraction: the denominator $1 + \frac{1}{1+\cdots}$ equals $x$, so write $x = 1 + \frac{1}{x}$. You now have a solvable equation.

For radicals, square both sides to remove the square root: $x^2 = 2 + x$. For continued fractions, multiply both sides by the denominator: multiplying $x = 1 + \frac{1}{x}$ by $x$ gives $x^2 = x + 1$. Both procedures produce a quadratic equation of the form $ax^2 + bx + c = 0$.

Rearrange and apply the quadratic formula or factor. For $x^2 - x - 2 = 0$: factor as $(x-2)(x+1) = 0$, giving $x = 2$ or $x = -1$. For $x^2 - x - 1 = 0$: the quadratic formula gives $x = \frac{1 \pm \sqrt{5}}{2}$.

Because square roots and most continued fractions represent positive quantities, you must check which root is physically valid. For $x = \sqrt{2+x}$, only $x \geq 0$ is acceptable, so $x = 2$ (reject $x = -1$). For $x = 1 + \frac{1}{x}$, we need $x > 0$, so $x = \frac{1+\sqrt{5}}{2} \approx 1.618$ — the famous golden ratio $\varphi$ (reject the negative root). Always verify your answer by substitution.

Plug your answer back into the original equation. For $x = 2$: $\sqrt{2+2} = \sqrt{4} = 2$ ✓. For $x = \varphi = \frac{1+\sqrt{5}}{2}$: $1 + \frac{1}{\varphi} = 1 + \frac{2}{1+\sqrt{5}} = 1 + \frac{1+\sqrt{5}-1+1}{(\ldots)}$. More directly, $\varphi^2 = \varphi + 1$ implies $\frac{\varphi^2 - 1}{\varphi} = 1$, which confirms $1 + \frac{1}{\varphi} = \varphi$ ✓.

• x — Value of the nested radical (dimensionless): The number that the infinite nested radical converges to; must be non-negative.
• a — Constant term (dimensionless): The fixed number added under each radical; must satisfy a > -1/4 for real solutions to exist.
• \frac{1+\sqrt{1+4a}}{2} — Positive root (dimensionless): The unique non-negative solution, obtained from the quadratic formula with the positive square root sign chosen.

Problem. Find the exact value of the infinite nested fraction: $x = \cfrac{3}{1 + \cfrac{3}{1 + \cfrac{3}{1 + \cdots}}}$

Answer. x = (−1 + √13) / 2 ≈ 1.303

• The Quadratic Formula — After reducing a nested radical or fraction to a polynomial equation, the quadratic formula is the primary tool for finding exact solutions.
• Radical Equations — Nested radicals are an advanced form of radical equation; understanding how to square both sides and check for extraneous solutions is essential prerequisite knowledge.
• The Golden Ratio — The golden ratio φ = (1+√5)/2 is the canonical example of a value defined by an infinite continued fraction, making it a direct application of the nested fraction technique.
• Infinite Sequences and Series — The convergence of an infinite nested expression to a finite value is rigorously justified using the theory of sequences, connecting to the broader study of limits.
• Extraneous Solutions — Squaring both sides of a nested radical equation can introduce false solutions; understanding why extraneous roots appear and how to eliminate them is critical to this topic.