### 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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