Easy Nested Functions

Algebra Level 3

625 + 625 + 26 625 + 51 625 + = ? \sqrt{625+\sqrt{625+26\sqrt{625+51\sqrt{625+\cdots}}}}=?


The answer is 26.

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1 solution

Chew-Seong Cheong
Mar 31, 2018

Using the Ramanujan's formula for nested radical (26) :

x + n + a = a x + ( n + a ) 2 + x a ( x + n ) + ( n + a ) 2 + ( x + n ) a ( x + 2 n ) + ( n + a ) 2 + ( x + 2 n ) x+n+a = \sqrt{ax + (n+a)^2 + x\sqrt{a(x+n) + (n+a)^2 + (x+n)\sqrt{a(x+2n) + (n+a)^2 + (x+2n)\sqrt{\cdots}}}}

and putting x = 1 x=1 , a = 0 a=0 and n = 25 n=25 , we have 1 + 0 + 25 = 625 + 625 + 26 625 + 51 = 26 1+0+25 = \sqrt{625+\sqrt{625+26\sqrt{625+51\sqrt{\cdots}}}} = \boxed{26} .

Yeah !! I also think of same however I'm trying it doing differently. :)

Naren Bhandari - 3 years, 2 months ago

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