The Root of the Problem The Third

Algebra Level 4

Solve for x x :

x = 9 + 2 9 + 5 9 + 8 9 + 11 9 + ⋯ \large x = \sqrt{9 + 2\sqrt{9 + 5\sqrt{9 + 8\sqrt{9 + 11\sqrt{9 +\cdots }}}}}


The answer is 5.

Deva Craig by Deva Craig , 22, USA

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1 solution

Ramanujan's formula (eqn. 26) on nested radical:

u + n + a = a u + ( n + a ) 2 + u a ( u + n ) + ( n + a ) 2 + ( u + n ) a ( u + 2 n ) + ( n + a ) 2 + ( u + 2 n ) ⋯ u+n+a = \sqrt{au+(n+a)^2 + u\sqrt{a(u+n)+(n+a)^2 + (u+n)\sqrt{a(u+2n)+(n+a)^2 + (u+2n)\sqrt \cdots}}}

Putting u = 2 u=2 , n = 3 n =3 and a = 0 a=0 , we have:

2 + 3 + 0 = 9 + 2 9 + 5 9 + 8 9 + 11 9 + ⋯ ⟹ x = 5 \begin{aligned} 2+3+0 & = \sqrt{9 + 2\sqrt{9 + 5\sqrt{9 + 8 \sqrt{9+11\sqrt {9+\cdots}}}}} \\ \implies x & = \boxed{5} \end{aligned}

11 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice question and an informative solution learned something new today. Thanks and keep posting such questions

ABHIJIT DIXIT - 4 years ago

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