Eudoxus' ladder

Eudoxus of Cnidus (now the tip of SW Turkey) was a renowned mathematician (amongst other things) who lived around 400 BC

one of the methods he (reportedly) developed or adopted when studying irrationals was to use a number series (called his 'ladder') in order to approximate to the square root of 2:

how is the 'ladder' formed?
how can it be used to approximate to the square root of 2?









'Reaching the Core of AS Mathematics', available from the ATM, interestingly links this blended recursion 'ladder' to the expansion of:

work out and simplify this expression for n = 2, 3, 4 etc

what has it got to do with Edoxus' ladder and why?

in the limit,

how does this provide an approximation to the square root of 2?