Continued root problem

Algebra Level 4

Find the value of the continued root

√(4 + 27√(4 + 29√(4 + 31√(4 + 33√.....


The answer is 29.

Arka Dutta by Arka Dutta , 19, India

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1 solution

a = 4 + ( a 2 − 4 ) = 4 + ( a − 2 ) ( a + 2 ) = 4 + ( a − 2 ) ( a + 2 ) 2 = 4 + ( a − 2 ) 4 + ( ( a + 2 ) 2 − 4 ) = 4 + ( a − 2 ) 4 + a ( a + 4 ) = 4 + ( a − 2 ) 4 + a ( a + 4 ) 2 = 4 + ( a − 2 ) 4 + a 4 + ( a + 2 ) 4 + ( a + 4 ) ⋯ The given expression is, = 4 + 27 4 + 29 4 + 31 4 + 33 ⋯ On comparing we get, ( a − 2 ) = 27 ⟹ a = 29 \begin{aligned}a&=\sqrt{4+(a^2-4)}\\ &=\sqrt{4+(a-2)(a+2)}\\ &=\sqrt{4+(a-2)\sqrt{(a+2)^2}}\\ &=\sqrt{4+(a-2)\sqrt{4+((a+2)^2-4)}}\\ &=\sqrt{4+(a-2)\sqrt{4+a(a+4)}}\\ &=\sqrt{4+(a-2)\sqrt{4+a \sqrt{(a+4)^2}}}\\ &=\sqrt{4+(a-2)\sqrt{4+a \sqrt{4+(a+2)\sqrt{4+(a+4)\sqrt{\cdots}}}}}\\\\ &\text{The given expression is,}\\ &=\sqrt{4+27\sqrt{4+29 \sqrt{4+31\sqrt{4+33\sqrt{\cdots}}}}}\\\\ &\text{On comparing we get,}\\ (a-2)&=27\\ \implies a&=\color{#EC7300}\boxed{\color{#333333}29}\end{aligned}

2 Helpful 2 Interesting 3 Brilliant 2 Confused

Fantastic way to solve. U have thoroughly generalized the equation.

Arka Dutta - 2 years, 2 months ago

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I did the same way........ @Arka Dutta What was your approach???

Aaghaz Mahajan - 2 years ago

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