Evaluate the sum using the summation formula and property.
$\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$
A)89,150
B)88,950
C)89,250
D)88,850
E)89,050

Question

Evaluate the sum using the summation formula and property. 
 $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$  
A)89,150
B)88,950
C)89,250
D)88,850
E)89,050

Quizplus · Accepted Answer

The given sum can be split into two separate sums using the summation property: $\sum _ { j = 1 } ^ { 25 } j ^ { 3 } - \sum _ { j = 1 } ^ { 25 } 3 j ^ { 2 }$. The formula for the sum of cubes is $\frac{n^2(n+1)^2}{4}$ and for the sum of squares is $\frac{n(n+1)(2n+1)}{6}$. Substituting $n=25$, we get $\frac{25^2(26)^2}{4} - 3\left(\frac{25(26)(51)}{6}\right)$, which simplifies to 89,050.

Evaluate the Sum Using the Summation Formula and Property ∑j=125(j3−3j2)\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)∑j=125​(j3−3j2)

Evaluate the Sum Using the Summation Formula and Property $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$