Find the sum using the formulas for the sums of powers of integers. $$\sum _ { n = 1 } ^ { 8 } \left( 7 n - 3 n ^ { 2 } \right)$$
A)-224
B)-1080
C)-136
D)144
E)-360

Question

Quizplus · Accepted Answer

The sum can be split into two separate sums: $\sum_{n=1}^{8} 7n - \sum_{n=1}^{8} 3n^2$. The first sum is an arithmetic series, and the second is a sum of squares. Using the formulas for these, $\sum_{n=1}^{8} n = \frac{8(8+1)}{2} = 36$ and $\sum_{n=1}^{8} n^2 = \frac{8(8+1)(2 \cdot 8 + 1)}{6} = 204$. Thus, the sum is $7 \cdot 36 - 3 \cdot 204 = 252 - 612 = -360$.

Find the Sum Using the Formulas for the Sums of Powers