İç İçe Sonsuz Kökler
\( a) \) $$ \bullet \quad \large \sqrt[n]{ a\sqrt[n]{ a\sqrt[n]{ a\sqrt[n]{ a \cdots } } } } = \sqrt[n-1]{ a} $$
\( \large \sqrt[n]{a \underbrace{\sqrt[n]{a \sqrt[n]{a \sqrt[n]{a \cdots}}}}_{x}} = \sqrt[n-1]{a} \Rightarrow \sqrt[n]{ a. x } = x \)
\( \large x = \sqrt[n-1]{a } \)
şeklinde doğruluğu gösterilebilir.
Örnekler:
\( \bullet \quad \sqrt[4]{ 8\sqrt[4]{ 8\sqrt[4]{ 8} \cdots }} = \sqrt[4-1]{ 8} = 2 \)
\( \bullet \quad \sqrt{ 7\sqrt{ 7\sqrt{ 7} \cdots }} = \sqrt[2-1]{ 7} = 7 \)
Soru 1
$$ \sqrt[4]{ 3\sqrt{ 3\sqrt[4]{ 8 \sqrt{3 \cdots } } }} $$ ifadesinin eşiti nedir?
\[
\text{A) } \sqrt[7]{ 27} \quad
\text{B) } \sqrt[7]{ 9} \quad
\text{C) } \sqrt[7]{ 3} \quad
\text{D) } 3 \quad
\text{E) } 1
\]
Çözüm
$$ \sqrt[4]{ 3\sqrt{ 3\sqrt[4]{ 8 \sqrt{3 \cdots } } }} = \sqrt[4]{ \sqrt{3 \cdot 3^2 \sqrt[4]{ \sqrt{ 3 \cdot 3^2} \cdots } } }=\sqrt[8]{27 \sqrt[8]{27 \cdots } } $$
$$ \sqrt[8-1]{27 } = \sqrt[7]{27 } $$
\(\textbf{Cevab: A} \)
\( b) \) $$ \bullet \quad \large \sqrt[n]{ a: \sqrt[n]{ a: \sqrt[n]{ a: \sqrt[n]{ a : \cdots } } } } = \sqrt[n+1]{ a} $$
\( \large \sqrt[n]{a : \underbrace{\sqrt[n]{a : \sqrt[n]{a: \sqrt[n]{a :\cdots}}}}_{x}} = x \Rightarrow \sqrt[n]{ a: x } = x \)
\( \large x = \sqrt[n+1]{ a} \)
şeklinde doğruluğu gösterilebilir.
Örnekler:
\( \bullet \quad \sqrt{ 2 : \sqrt{ 2 : \sqrt { 2: } \cdots }} = \sqrt[2+1]{ 8} = 2 \)
Soru 2
$$ \frac{\sqrt{ 3: \sqrt{ 3 : \sqrt{ 3 \cdots } } } }{ \sqrt[4]{ 3 : \sqrt[4]{3 : \sqrt[4]{3 \cdots } } } } $$ ifadesinin eşiti nedir?
\[
\text{A) } 9 \quad
\text{B) } 3 \quad
\text{C) } 1 \quad
\text{D) } \frac{1}{3} \quad
\text{E) } \frac{1}{9}
\]
Çözüm
$$ \frac{\sqrt{ 3: \sqrt{ 3 : \sqrt{ 3 \cdots } } } }{ \sqrt[4]{ 3 : \sqrt[4]{3 : \sqrt[4]{3 \cdots } } } } = \frac{\sqrt[2+1]{ 3} }{\sqrt[4-1]{ 3} } = \frac{\sqrt[3]{ 3} }{\sqrt[3]{3 } } = 1 $$
\(\textbf{Cevab: C} \)
\( c) \) $$ \bullet \quad \large \sqrt[n]{ a+ \sqrt[n]{ a+ \sqrt[n]{ a+ \sqrt[n]{ a + \cdots } } } } = \frac{1 \; + \; \sqrt{ 1 \;+ \; 4a} }{2} \quad (a \;> \;0)$$
$$ \bullet \quad \large \sqrt[n]{ a- \sqrt[n]{ a- \sqrt[n]{ a- \sqrt[n]{ a – \cdots } } } } = \frac{ -1 \; + \; \sqrt{ 1 \;+ \; 4a} }{2} \quad (a \;≥ \; 0)$$
\[ \sqrt{a \pm \underbrace{\sqrt{a \pm \sqrt{a \pm \cdots}}}_{x} }= x \implies \sqrt{a \pm x} = x \] \[ \implies a \pm x = x^2 \] \[ \implies x = \frac{\pm 1 + \sqrt{1 + 4a}}{2} \]
şeklinde doğruluğu gösterilebilir.
Örnekler:
\( \bullet \quad \sqrt{ 5 +\underbrace{ \sqrt{ 5 +\sqrt { 5+ } \cdots }}_{x}} = x \Rightarrow \sqrt{5+x } =x \Rightarrow 5 + x = x^2 \)
$$ \Rightarrow x^2 \;- \; x \;- \; 5= 0 $$
$$ \Rightarrow x = \frac{1 \; + \; \sqrt{ 1 + 4 \cdot 5} }{2} $$
$$ \Rightarrow x = \frac{1 \; + \; \sqrt{ 21 }}{2} $$
Soru 3
$$ \sqrt{ x – \sqrt{ x -\sqrt x- \cdots }}= 2 $$ ise x kaçtır?
\[
\text{A) } 3 \quad
\text{B) } 4 \quad
\text{C) } 5 \quad
\text{D) } 6 \quad
\text{E) } 7
\]
Çözüm
$$ \sqrt{ x – \underbrace{ \sqrt{ x -\sqrt x- \cdots }}_{2}}= 2 $$
$$ \Rightarrow \sqrt{x-2 } = 2 $$
$$ \Rightarrow x-2 = 2^2 $$
$$ \Rightarrow x= 6 $$
\(\textbf{Cevab: D} \)
Uyarı:
$$a > 0 \quad \text{olmak üzere} $$
$$ \sqrt{ a(a\;+ \;1) + \sqrt{ a(a\;+\;1) + \sqrt{ a(a\;+\;1 ) \cdots} } } = a\; +\;1 $$
$$ \sqrt{ a(a\;- \;1) + \sqrt{ a(a\;-\;1) + \sqrt{ a(a\;-\;1 ) \cdots} } } = a$$
Örnekler:
\( \bullet \quad \sqrt{\underbrace{12}_{3 . 4 } + \sqrt{ 12+ \sqrt{12 } } \cdots } = 4 \quad (a = 3, \; a+1 = 4) \)
\( \bullet \quad \sqrt{\underbrace{30}_{5 . 6 } – \sqrt{ 30- \sqrt{30 } } \cdots } = 5 \quad (a = 5, \; a+1 = 6) \)
Soru 4
$$ \sqrt{ 7- \sqrt{6 + \sqrt{6 + \sqrt{6 } } \cdots } } \:+ \; \sqrt{ 2- \sqrt{2 – \sqrt{2 – \sqrt{2 }} } \cdots } $$ işleminin sonucu kaçtır?
\[
\text{A) } 1 \quad
\text{B) } 2 \quad
\text{C) } 3 \quad
\text{D) } 4 \quad
\text{E) } 5
\]
Çözüm
$$ =\sqrt{ 7- \underbrace{\sqrt{2. 3 + \sqrt{2.3 + \sqrt{2. 3 } } \cdots } }_{3} } \:+ \; \underbrace{ \sqrt{ 1.2- \sqrt{1.2 – \sqrt{1.2 – \sqrt{1. 2 }} } \cdots } }_{1}$$
$$ \Rightarrow \sqrt{7-3 } + 1 = 3$$
\(\textbf{Cevab: C} \)
Soru 5
$$ \sqrt{ \frac{3}{4} + \sqrt{\frac{3}{4} + \sqrt{\frac{3}{4} + \sqrt{\frac{3}{4} } } \cdots } } $$ işleminin sonucu kaçtır?
\[
\text{A) } \frac{1}{2} \quad
\text{B) } 1 \quad
\text{C) } \frac{3}{2} \quad
\text{D) } 2 \quad
\text{E) } \frac{5}{2}
\]
Çözüm
$$ \sqrt{ \underbrace{\frac{3}{4}}_{\frac{1}{2} \cdot (\frac{1}{2}+1 )} + \sqrt{\frac{3}{4} + \sqrt{\frac{3}{4} + \sqrt{\frac{3}{4} } } \cdots } } $$
$$ \frac{3}{4}= \frac{1}{2} \cdot ( \frac{1}{2} +1) $$
$$ \frac{1}{2} +1 = \frac{3}{2} $$
\(\textbf{Cevab: C} \)
Soru 5
$$ \sqrt[3]{x+1 + \sqrt[3]{x+1 + \sqrt[3]{x+1 } \cdots } } = 2 $$ ise x kaçtır?
\[
\text{A) } 1 \quad
\text{B) } 2\quad
\text{C) } 3 \quad
\text{D) } 4 \quad
\text{E) } 5
\]
Çözüm
$$ \sqrt[3]{x+1 + \underbrace{ \sqrt[3]{x+1 + \sqrt[3]{x+1 } \cdots } }_{2} } = 2 $$
$$ \sqrt[3]{ x\;+\;1 \;+\;2 } = 2 \Rightarrow (\sqrt[3]{ \;x\;+\;1 \;+\;2 } )^3\;= \;2^3 $$
$$ \Rightarrow \;x \;+\;3\; = \;8 $$
$$ \Rightarrow x \; =\; 5 $$
\(\textbf{Cevab: E} \)