Processing math: 100%

Wednesday, December 14, 2022

A surprising expression with nested square roots

 Simplify 


10+1+10+2++10+99101+102++1099


Scroll down for simplified form and solution.




The expression is equal to 2+1.  Surprising!


Scroll down for solution.



Let a+b = 100. 


Now

10a10+a=100a=b


Let u=10+a and v=10a

So 20=u2+v2

and b=uv

Thus

20+2b=u2+v2+2uv=u+v

Similarly

202b=u2+v22uv=uv

Thus if 

P=99a=110+a

and

Q=99a=110a

We get 
2P=P+Q
and

2Q=PQ

This gives us PQ=2+1

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