In our proof that repeating numbers are rational, we used the following relationship:

S = Sum of the geometric series ar

^{n-1}

= a + ar

^{1}+ ar

^{2}+ ar

^{3}+ ar

^{4}......

= a/(1-r)

Where does this rather elegant and surprising relationship come from ? Certainly, this simple realtionship is rather unexpected .... why would an infinite series converge on this simple ratio ?

Like many relationships in mathematics, the proof is beautifully simple. Firstly, form the equation S - Sr = a + ar

^{1}+ ar

^{2}+ ar

^{3}..... - ar

^{1}+ ar

^{2}+ ar

^{3}.... = a

Therefore, rearranging we arrive at S = a/(1-r).

QED

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