Technique

Value of an infinite nested radical

Algebra • Surds • Nested Radicals

Statement

x = a + a + a + \dots

with

a > 0

, the expression inside the first root is the whole expression again, so

x^{2} = a + x

and

x = \frac{1 + 1 + 4 a}{2}

; the same move on

x = a a a \dots

gives

x^{2} = a x

How to use it

1Name the whole expression $x$ , then find the identical copy of $x$ sitting inside it.
2Square once to clear the outer radical and solve the quadratic that appears.
3Reject the negative root: the radical sign denotes the non-negative value.

Worked example

Find the value of

6 + 6 + 6 + \dots

Step-by-step

1
$x = 6 + x$
Why: What sits under the first root is the whole expression over again.
2
$x^{2} - x - 6 = 0 \Rightarrow (x - 3) (x + 2) = 0$
3
$x = 3$
Why: The negative root is discarded because $x$ is a square root.

Answer

3

Watch out

The substitution assumes the radical converges; for

a > 0

the finite truncations increase and stay bounded, so it does, but a nest of finitely many roots is a smaller number and must not be evaluated this way.

Builds on

denotes the non-negative root Quadratic formula

Also known as

infinite nested radical, self-similar radical, continued radical, square root of 1 plus square root of 1, fixed point of a radical

Value of an infinite nested radical

Statement

Worked example

Appears in past papers

2019 (I)

2017 (II)