Answer
$3,251,250$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given expression, $
\displaystyle\sum_{i=1}^{50} 2i^3
,$ use the properties of summation.
$\bf{\text{Solution Details:}}$
Using a property of the summation which is given by $\displaystyle\sum_{i=1}^n ca_i=c\displaystyle\sum_{i=1}^n a_i,$ with $c=
2
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
2\displaystyle\sum_{i=1}^{50} i^3
.\end{array}
Using a property of the summation which is given by $\displaystyle\sum_{i=1}^n i^3=\dfrac{n^2(n+1)^2}{4},$ with $n=
50
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
2\left( \dfrac{50^2(50+1)^2}{4} \right)
\\\\=
2\left( \dfrac{50^2(51)^2}{4} \right)
\\\\=
3,251,250
.\end{array}
