Find the value of \(\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\ldots}}}}\)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 3
The roots go on for ever, so the part under the first root sign, after “6 +”, is the whole expression again. Calling the value \(x\) gives an equation in \(x\):
\(x=\sqrt{6+x}\)
\(x^{2}=6+x\)
\(x^{2}-x-6=0\)
\((x-3)(x+2)=0\), so \(x=3\) or \(x=-2\)
A square root cannot be negative, so \(x=3\)