### Brain Teasers

# Square Root of 2

An iterative equation is one where an initial value is put into the equation to obtain a result, which is in turn put back into the equation as an input and the process is repeated for either a fixed number of times or until a final value is achieved (where you get the same number out as you put in).

For example, if you start with the iterative equation,

A{n+1} = 1/A{n} + 1

And you put in an initial value of 1, you would get 2 as the result. Put the 2 back in as the input and you would get 1.5, and so on.

There is a simple iterative equation which, when repeated just five times, will give you a very good approximation of the square root of two (accurate to three decimal places, or less than 0.1%). What's more, you can put in any positive (and almost any negative) initial value into the equation (even zero) and you will end up at the same answer. This equation does not contain any square root expressions; just single-order algebra. Can you figure it out?

For example, if you start with the iterative equation,

A{n+1} = 1/A{n} + 1

And you put in an initial value of 1, you would get 2 as the result. Put the 2 back in as the input and you would get 1.5, and so on.

There is a simple iterative equation which, when repeated just five times, will give you a very good approximation of the square root of two (accurate to three decimal places, or less than 0.1%). What's more, you can put in any positive (and almost any negative) initial value into the equation (even zero) and you will end up at the same answer. This equation does not contain any square root expressions; just single-order algebra. Can you figure it out?

### Hint

The only mathematical operators you need are addition and division. Try beginning with the expression:A^2 = 2

And keep in mind that this equation converges on a value. Don't give up!

### Answer

A{n+1} = (A{n} + 2) / (A{n} + 1}To prove to yourself that this is right, assume that for a sufficiently large number of iterations, A{n+1} = A{n}, so you can rewrite the equation as:

A(A+1) = A + 2

... A^2 + A = A + 2

... A^2 = 2

... A = SQRT(2) !

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