# Task 4.

## Maximum 15 points

Find the value of the expression under the condition that the summation is performed infinitely

$$
\sqrt{20+\sqrt{20+\sqrt{20+\cdots}}}
$$

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# Solution:

Let $x=\sqrt{20+\sqrt{20+\sqrt{20+\cdots}}}, x>0$.

Square both sides of the obtained equation. We get:

$x^{2}=20+x$

$x^{2}-x-20=0 ;$

$D=1+80=81 ; x_{1}=\frac{1-9}{2}=-4-$ does not satisfy the condition $x>0$;

$x_{1}=\frac{1+9}{2}=5-$ fits.

Answer: 5.

## Criteria

0 points - If calculated approximately, then $0 \%$.

8 points - If both roots of the equation are given: -4 and 5, then $50 \%$.

15 points - If the answer is correct, and a valid and justified solution is provided without using a calculator, then $100 \%$.

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