Perfect Infinity

Level 1

If 1 + 1 + 1 + 1 + 1 + 1 + 1... \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1...}}}}}}} can be written as A + B C \frac{A+\sqrt{B}}{C} , where A A , B B , and C C are coprime positive integers, find A + B + C A+B+C .


The answer is 8.

Daniel Xian by Daniel Xian , 15, New Zealand

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Joshua Lowrance
Jan 25, 2019

X = 1 + 1 + 1 + 1 + 1 + 1 + 1 ⋯ X=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1\cdots}}}}}}}

X = 1 + 1 + 1 + 1 + 1 + 1 + 1 ⋯ X=\sqrt{1+\color{#3D99F6} \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1\cdots}}}}}}}

X = 1 + X X=\sqrt{1+X}

X 2 = 1 + X X^2=1+X

X 2 − X − 1 = 0 X^2-X-1=0

Here, you can solve the quadratic, or just know that the answer to this is phi = 1.618... = 1 + 5 2 =1.618... = \frac{1+\sqrt{5}}{2} .

3 Helpful 2 Interesting 1 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...