Approximation to Golden Rectangle using Fibonacci Squares/Proof 1

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An approximation to a golden rectangle can be obtained by placing adjacent to one another squares with side lengths corresponding to consecutive Fibonacci numbers in the following manner:


It can also be noted, as from Sequence of Golden Rectangles, that an equiangular spiral can be approximated by constructing quarter circles as indicated.


Let the last two squares to be added have side lengths of $F_{n - 1}$ and $F_n$.

Then from the method of construction, the sides of the rectangle generated will be $F_n$ and $F_{n + 1}$.

From Continued Fraction Expansion of Golden Mean it follows that the limit of the ratio of the side lengths of the rectangle, as $n$ tends to infinity, is the golden section $\phi$.

Hence the result.