Find the sum of n-terms of …

Let {tex}{{\rm{S}}_n} = \sum {{{\left( {{\rm{odd\ number}}} \right)}^3}} {/tex}

=>     {tex}{{\rm{S}}_n} = \sum\limits_{k = 1}^n {\left( {2k - 1} \right)} {/tex}                      [S_n = Sum of n odd natural numbers]

=>     {tex}{{\rm{S}}_n} = \sum\limits_{k = 1}^n {\left( {8{k^3} - 12{k^2} + 6x - 1} \right)} {/tex}

=>     {tex}{{\rm{S}}_n} = 8\left( {\sum\limits_{k = 1}^n {{k^3}} } \right) - 12\left( {\sum\limits_{k = 1}^n {{k^3}} } \right) + 6\left( {\sum\limits_{k = 1}^n k } \right) - n{/tex}       [1 + 1 + 1 + ...... to n terms = n]

=>     {tex}{{\rm{S}}_n} = 8.{1 \over 4}{n^2}{\left( {n + 1} \right)^2} - 12.{1 \over 6}n\left( {n + 1} \right)\left( {2n + 1} \right) + 6.{1 \over 2}n\left( {n + 1} \right) - n{/tex}

=>     {tex}{{\rm{S}}_n} = 2{n^2}{\left( {n + 1} \right)^2} - 2n\left( {n + 1} \right)\left( {2n + 1} \right) + 3n\left( {n + 1} \right) - n{/tex}

=>     {tex}{{\rm{S}}_n} = n\left( {n + 1} \right)\left[ {2n\left( {n + 1} \right) - 2\left( {2n + 1} \right) + 3} \right] - n{/tex}

=>     {tex}{{\rm{S}}_n} = n\left( {n + 1} \right)\left[ {2{n^2} - 2n + 1} \right] - n{/tex}

=>     {tex}{{\rm{S}}_n} = n\left[ {\left( {n + 1} \right)\left( {2{n^2} - 2n + 1} \right) - 1} \right]{/tex}

=>     {tex}{{\rm{S}}_n} = n\left[ {2{n^3} - 2{n^2} + n + 2{n^2} - 2n + 1 - 1} \right]{/tex}

=>     {tex}{{\rm{S}}_n} = n\left[ {2{n^3} - n} \right]{/tex}

=>     {tex}{{\rm{S}}_n} = {n^2}\left[ {2{n^2} - 1} \right]{/tex}