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Prove that $3=\sqrt{1+2\sqrt{1+3 \sqrt{1+4 \sqrt{1+ \cdots}}}}.$

Best Answer 

 #2
avatar+26213 
+3

Prove that \(3=\sqrt{1+2\sqrt{1+3 \sqrt{1+4 \sqrt{1+ \cdots}}}}\).

 

Source: https://www.quora.com/What-is-the-solution-of-sqrt-1+2-sqrt-1+3-sqrt-1+4-sqrt

 

\(\small{ \begin{array}{|rcll|} \hline n(n+2) &=& n\sqrt{(n+2)^2} \\ &=& n\sqrt{n^2+4n+4} \\ &=& n\sqrt{1+n^2+4n+3} \\ &=& n\sqrt{1+n^2+3n+n+3} \\ \mathbf{ n(n+2) } &=& \mathbf{ n\sqrt{1+(n+1)(n+3)} } \\ \hline f(n) &=& n(n+2) \\\\ f(n+1) &=& (n+1)(n+1+2 ) \\ f(n+1) &=& (n+1)(n+3) \\ \mathbf{ n(n+2) } &=& \mathbf{ n\sqrt{1+(n+1)(n+3)} } \\ f(n) &=& n\sqrt{1+f(n+1)} \\ && \boxed{f(n+1) = (n+1)\sqrt{1+f(n+1+1)}\\ f(n+1) = (n+1)\sqrt{1+f(n+2)} } \\ f(n) &=& n\sqrt{1+(n+1)\sqrt{1+f(n+2)}} \\ && \boxed{f(n+2) = (n+2)\sqrt{1+f(n+2+1)}\\ f(n+2) = (n+2)\sqrt{1+f(n+3)} } \\ f(n) &=& n\sqrt{1+(n+1)\sqrt{1+ (n+2)\sqrt{1+f(n+3)}}} \\ && \boxed{f(n+3) = (n+3)\sqrt{1+f(n+3+1)}\\ f(n+3) = (n+3)\sqrt{1+f(n+4)} } \\ f(n) &=& n\sqrt{1+(n+1)\sqrt{1+(n+2)\sqrt{1+(n+3)\sqrt{1+f(n+4)}}}} \quad | \quad f(n) = n(n+2)\\ n(n+2) &=& n\sqrt{1+(n+1)\sqrt{1+(n+2)\sqrt{1+(n+3)\sqrt{1+f(n+4)}}}} \quad | \quad n = 1 \\ 1(1+2) &=& 1*\sqrt{1+(1+1)\sqrt{1+(1+2)\sqrt{1+(1+3)\sqrt{1+\dots}}}} \\ 3 &=& \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\dots}}}} \\ \hline \end{array} }\)

 

laugh

 Jun 2, 2021
edited by heureka  Jun 2, 2021
 #1
avatar
+1

Wow! That sure is an interesting question. I plugged the first numbers into my calculator and it came close to 3, so I think it is correct, but "It works because my calculator said so" isn't a good proof (unfortunately).

 

I do know that we can solve for x when we have: \(x = \sqrt{y+\sqrt{y+\sqrt{y+...}}}\) because we can square both sides, subtract by y, and then the RHS of the equation is x, so we are left with \(x^{2}-y=x\).

 

However, there is no reoccurring pattern in your problem, it is just an ongoing arithmetic sequence. So I'm not really not sure how to find it. However, if you set the RHS equal to x, square both sides of your equation, and then play with it to get the RHS of your equation to come up again, then you can solve it.

 Jun 2, 2021
 #2
avatar+26213 
+3
Best Answer

Prove that \(3=\sqrt{1+2\sqrt{1+3 \sqrt{1+4 \sqrt{1+ \cdots}}}}\).

 

Source: https://www.quora.com/What-is-the-solution-of-sqrt-1+2-sqrt-1+3-sqrt-1+4-sqrt

 

\(\small{ \begin{array}{|rcll|} \hline n(n+2) &=& n\sqrt{(n+2)^2} \\ &=& n\sqrt{n^2+4n+4} \\ &=& n\sqrt{1+n^2+4n+3} \\ &=& n\sqrt{1+n^2+3n+n+3} \\ \mathbf{ n(n+2) } &=& \mathbf{ n\sqrt{1+(n+1)(n+3)} } \\ \hline f(n) &=& n(n+2) \\\\ f(n+1) &=& (n+1)(n+1+2 ) \\ f(n+1) &=& (n+1)(n+3) \\ \mathbf{ n(n+2) } &=& \mathbf{ n\sqrt{1+(n+1)(n+3)} } \\ f(n) &=& n\sqrt{1+f(n+1)} \\ && \boxed{f(n+1) = (n+1)\sqrt{1+f(n+1+1)}\\ f(n+1) = (n+1)\sqrt{1+f(n+2)} } \\ f(n) &=& n\sqrt{1+(n+1)\sqrt{1+f(n+2)}} \\ && \boxed{f(n+2) = (n+2)\sqrt{1+f(n+2+1)}\\ f(n+2) = (n+2)\sqrt{1+f(n+3)} } \\ f(n) &=& n\sqrt{1+(n+1)\sqrt{1+ (n+2)\sqrt{1+f(n+3)}}} \\ && \boxed{f(n+3) = (n+3)\sqrt{1+f(n+3+1)}\\ f(n+3) = (n+3)\sqrt{1+f(n+4)} } \\ f(n) &=& n\sqrt{1+(n+1)\sqrt{1+(n+2)\sqrt{1+(n+3)\sqrt{1+f(n+4)}}}} \quad | \quad f(n) = n(n+2)\\ n(n+2) &=& n\sqrt{1+(n+1)\sqrt{1+(n+2)\sqrt{1+(n+3)\sqrt{1+f(n+4)}}}} \quad | \quad n = 1 \\ 1(1+2) &=& 1*\sqrt{1+(1+1)\sqrt{1+(1+2)\sqrt{1+(1+3)\sqrt{1+\dots}}}} \\ 3 &=& \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\dots}}}} \\ \hline \end{array} }\)

 

laugh

heureka Jun 2, 2021
edited by heureka  Jun 2, 2021
 #3
avatar+121056 
0

Thx, heureka.......this  one  is  very difficult    !!!!

 

 

cool cool cool

CPhill  Jun 2, 2021
 #4
avatar+26213 
0

Thank you CPhill,

 

here is a nice youtube video

How To prove that 3=sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+...))))

with another  way: https://www.youtube.com/watch?v=9yMQMPj0zVU

 

laugh

heureka  Jun 2, 2021
edited by heureka  Jun 2, 2021
edited by heureka  Jun 2, 2021
 #5
avatar+26213 
0

Thank you CPhill,

 

here is a nice youtube video

How To prove that 3=sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+...))))

with another  way: https://www.youtube.com/watch?v=9yMQMPj0zVU

 

laugh

heureka  Jun 2, 2021

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