Find 1 + (1 + (1 + (1 + (1 + ...)^-1)^-1)^-1)^-1
1 + (1 + (1 + (1 + (1 + ...)-1)-1)-1)-1
This pattern goes infinitely.
The first step to this question, is to observe. Observe the pattern. Look at the brackets. Do you see how it repeats itself?
Let's look at the equation inside the first brackets: 1 + (1 + (1 + (1 + ...)-1)-1)-1 ______ (a)
And the next brackets: 1 + (1 + (1 + ...)-1)-1 . ______ (b)
Do (a) and (b) look similar? Try to expand the (b), we will get: 1 + (1 + (1 + (1 + ...)-1)-1)-1
YES! They are just the same equation! And compare (a) and (b) with the original question, they are the same!
Now we observed the pattern, then how do we solve it?
Let's say: x = 1 + (1 + (1 + (1 + (1 + ...)-1)-1)-1)-1
As we have already discovered, the equation inside the first bracket is the same as the whole equation, i.e. x.
So we can write this equation as: x = 1 + x-1
Now things are much easier! We simply multiply both sides with "x", then we get x2 = x + 1 => x2 - x - 1 = 0
Then we can solve this simple quadratic equation!
After solving the equation, you will find two roots to the equation: "(1 + sqrt(5)) / 2" and "(1 - sqrt(5)) / 2"
Which one is the correct answer to our original question?
To answer this question, you need to observe again.
Observe the two roots. 22 = 4 and 32 = 9Since 4 < 5 < 9, it is obvious that: 2 < sqrt(5) < 3.
So it is obvious that the root "(1 - sqrt(5)) / 2" is going to be less than 0. So we get one positive root and one negative root from our previous quadratic equation.
Look at our original question again, do you think that equation can ever go negative?
So which of the two roots is correct answer?
It is (1 + sqrt(5)) / 2!
And this is the famous: GOLDEN RATIO!!!!
