### Brain Teasers

# 1, 2, 1.5, 1.666...

What is the next number in this series:

1, 2, 1.5, 1.66666, 1.6, ?

1, 2, 1.5, 1.66666, 1.6, ?

### Answer

The next number is:1.625

The series constructed by, beginning with one, add one to the inverse of the previous number. If you keep going, you will eventually bottom out at:

1.618033988749895

Which is neat because that is one of the two solutions to the quadratic equation:

x^2 - x - 1 = 0

Which is the same as the equation:

x = 1/x + 1

Which is how each number in the series is generated, or,

x{n+1} = 1/x{n} + 1

Another interesting fact about this series: You will end up at the same number regardless of the positive number you begin with!

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### Solve a Puzzle

## Comments

Hint! This is an iterative function.

This number is the golden ratio. Even more interesting is the other solution to the quadratic which can be found by removing the 1 from the decimal and negating it. Try finding the reciprocal of each of these solutions!

Forget the quadratic formula. I solved it in a less dramatic way. I converted everything to improper fractions. Once you get past the first two numbers: the numerator of each number becomes the denominator of the following number, and the sum of the numerators of the last two numbers becomes the numerator of the following number. Then convert back to decimal.

yeah, i saw the Fibonacci sequence. Indeed, the Fibonacci sequence is related to the golden ratio and rectangle.

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