Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 22 } \left( n ^ { 3 } - n \right)$
A)22,770
B)3,795
C)63,756
D)256,036
E)64,009

Question

Find the sum using the formulas for the sums of powers of integers.   $\sum _ { n = 1 } ^ { 22 } \left( n ^ { 3 } - n \right)$  
A)22,770
B)3,795
C)63,756
D)256,036
E)64,009

Quizplus · Accepted Answer

The sum can be simplified using the formulas for the sums of powers of integers. The formula for the sum of cubes is $\frac{n^2(n+1)^2}{4}$ and for the sum of integers is $\frac{n(n+1)}{2}$. Thus, the sum is $\sum _ { n = 1 } ^ { 22 } n^3 - \sum _ { n = 1 } ^ { 22 } n = \frac{22^2(23)^2}{4} - \frac{22(23)}{2} = 63,756$.

Find the Sum Using the Formulas for the Sums of Powers