In our proof that repeating numbers are rational, we used the following relationship:
S = Sum of the geometric series arn-1
= a + ar1 + ar2 + ar3 + ar4......
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 + ar1 + ar2 + ar3 ..... - ar1 + ar2 + ar3 .... = a
Therefore, rearranging we arrive at S = a/(1-r).