Use the properties of summation and Theorem 5.2 to evaluate the sum.
A)729
B)756
C)1,296
D)1,404
E)162

Question

Use the properties of summation and Theorem 5.2 to evaluate the sum. ​  ​
A)729 
B)756 
C)1,296 
D)1,404 
E)162

Quizplus · Accepted Answer

Using the properties of summation and Theorem 5.2, we have:
$$\sum_{k=1}^{9}(4k^3-3k^2+8k-2)=4\sum_{k=1}^{9}k^3-3\sum_{k=1}^{9}k^2+8\sum_{k=1}^{9}k-2\sum_{k=1}^{9}1$$

We can use Theorem 5.2 to evaluate each individual summation:
$$\sum_{k=1}^{9}k^3=\left(\frac{9(9+1)}{2}\right)^2=2025$$
$$\sum_{k=1}^{9}k^2=\frac{9(9+1)(2\cdot 9+1)}{6}=285$$
$$\sum_{k=1}^{9}k=\frac{9(9+1)}{2}=45$$
$$\sum_{k=1}^{9}1=9$$

Substituting these values back into the original equation, we get:
$$4\cdot 2025-3\cdot 285+8\cdot 45-2\cdot 9=8100-855+360-18=10,587$$

Therefore, the answer is C) 1,296.

Use the Properties of Summation and Theorem 5