$1 + \frac{4}{3} + \frac{{10}}{9} + \frac{{28}}{{27}} + .....n$ terms
$ = 1 + \left( {1 + \frac{1}{3}} \right) + \left( {1 + \frac{1}{9}} \right) + \left( {1 + \frac{1}{{27}}} \right) + ....n$ terms
$ = \left( {1 + 1 + 1 + ..... + n\,\,terms} \right) + \left( {\frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + .....n\,\,terms} \right)$
$ = n + \frac{{\frac{1}{3}\left( {1 + \frac{1}{{{3^n}}}} \right)}}{{1 - \frac{1}{3}}} = n + \frac{1}{3} \times \frac{3}{2}\left[ {1 - {3^{ - n}}} \right]$
$ = n + \frac{1}{2}\left[ {1 - {3^{ - n}}} \right] = n + \frac{1}{2} - \frac{1}{{{{2.3}^n}}}$
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