Answer
$973$
Work Step by Step
$\sum ^{6}_{i=1}\left( i^{2}+2i^{3}\right) =\sum ^{6}_{i=1}i^{2}+\sum ^{6}_{i=1}2i^{3}=\dfrac {n\left( n+1\right) \left( 2n+1\right) }{6}+\dfrac {2\times n^{2}\left( n+1\right) ^{2}}{4}=\dfrac {6\times \left( 6+1\right) \left( 2\times 6+1\right) }{6}+\dfrac {2\times 6^{2}\times \left( 6+1\right) ^{2}}{4}=973 $
