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SpyIntel [72]
2 years ago
11

Please help.. (This is meant to be infinite, so I can't use a calculator)

Mathematics
2 answers:
Alex17521 [72]2 years ago
7 0

Answer:

5

Step-by-step explanation:

→ Let x = √20 + √20 .....

x

→ Square both sides

x² = 20 + √20 +√20 + √20

→ Replace √20 +√20 + √20 with x

x² = 20 + x

→ Move everything to the left hand side

x² - 20 - x = 0

→ Factorise

( x + 4 ) ( x - 5 )

→ Solve

x = -4 , 5

→ Discard negative result

x = 5

Elza [17]2 years ago
3 0

Note that

\left(\sqrt{x+a} + b\right)^2 = \left(\sqrt{x+a}\right)^2 + 2b \sqrt{x+a} + b^2 = x + 2b \sqrt{x+a} + a+b^2

Taking square roots on both sides, we have

\sqrt{x+a} + b = \sqrt{x + 2b \sqrt{x+a} + a+b^2}

Now suppose a+b^2=0 and 2b=1. Then b=\frac12 and a=-\frac14, so we can simply write

\sqrt{x-\dfrac14} + \dfrac12 = \sqrt{x + \sqrt{x - \dfrac14}}

This means

\sqrt{x-\dfrac14} = -\dfrac12 + \sqrt{x + \sqrt{x - \dfrac14}}

and substituting this into the root expression on the right gives

\sqrt{x - \dfrac14} + \dfrac12 = \sqrt{x - \dfrac12 + \sqrt{x + \sqrt{x - \dfrac14}}}

and again,

\sqrt{x - \dfrac14} + \dfrac12 = \sqrt{x - \dfrac12 + \sqrt{x - \dfrac12 + \sqrt{x - \dfrac14}}}

and again,

\sqrt{x - \dfrac14} + \dfrac12 = \sqrt{x - \dfrac12 + \sqrt{x - \dfrac12 + \sqrt{x - \dfrac12 + \sqrt{x - \dfrac14}}}}

and so on. After infinitely many iterations, the right side will converge to an infinitely nested root,

\sqrt{x - \dfrac14} + \dfrac12 = \sqrt{x - \dfrac12 + \sqrt{x - \dfrac12 + \sqrt{x - \dfrac12 + \sqrt{x - \dfrac12 + \sqrt{\cdots}}}}}

We get the target expression when x=\frac{41}2, since \frac{41}2-\frac12=\frac{40}2=20, which indicates the infinitely nested root converges to

\sqrt{20+\sqrt{20+\sqrt{20+\sqrt{\cdots}}}} = \sqrt{\dfrac{41}2 - \dfrac14} + \dfrac12 \\\\ ~~~~~~~~ = \sqrt{\dfrac{81}4} + \dfrac12 \\\\ ~~~~~~~~ = \dfrac92 + \dfrac12 = \dfrac{10}2 = \boxed{5}

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