The sum of first $n$ terms of the series $ \dfrac {4}{3} , \dfrac {10}{9} , \dfrac {28}{27} , \dfrac {244}{243} , ... $ is :

We can split the terms into whole numbers and fractional parts, writing the general term as (1 + 1/3^r) where r starts from 1. The sum of the first n terms is the sum of n ones plus the sum of the geometric series (1/3 + 1/9 + 1/27 + ... up to n terms). The geometric series sum equals (1/3)(1 - (1/3)^n) / (1 - 1/3), which simplifies to 1/2(1 - 3^{-n}). Adding the n ones gives the total sum as n + 1/2(1 - 3^{-n}).

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