Use the summation properties to evaluate the series. The following rules may be needed: $$\begin{array} { l l l }
\mathrm { n } & \mathrm { n } & \mathrm { n } \
\sum \mathrm { i } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) } { 2 } ; & \sum \mathrm { i } ^ { 2 } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) ( 2 \mathrm { n } + 1 ) } { 6 } ; & \sum \mathrm { i } ^ { 3 } = \frac { \mathrm { n } ^ { 2 } ( \mathrm { n } + 1 ) ^ { 2 } } { 4 } \
\mathrm { i } = 1 & \mathrm { i } = 1 & \mathrm { i } = 1
\end{array}$$
-$$\sum _ { i = 1 } ^ { 4 } \left( - 2 i ^ { 2 } + i \right)$$
A) $- 80$
B) $- 50$
C) 40
D) $- 70$

Question

Use the summation properties to evaluate the series. The following rules may be needed: $$\begin{array} { l l l } 
\mathrm { n } & \mathrm { n } & \mathrm { n } \
\sum \mathrm { i } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) } { 2 } ; & \sum \mathrm { i } ^ { 2 } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) ( 2 \mathrm { n } + 1 ) } { 6 } ; & \sum \mathrm { i } ^ { 3 } = \frac { \mathrm { n } ^ { 2 } ( \mathrm { n } + 1 ) ^ { 2 } } { 4 } \
\mathrm { i } = 1 & \mathrm { i } = 1 & \mathrm { i } = 1
\end{array}$$
-$$\sum _ { i = 1 } ^ { 4 } \left( - 2 i ^ { 2 } + i \right)$$
A) $- 80$
B) $- 50$
C) 40
D) $- 70$

Quizplus · Accepted Answer

To evaluate the series $\sum _ { i = 1 } ^ { 4 } \left( - 2 i ^ { 2 } + i \right)$, we can separate it into two parts using the given summation properties: $\sum _ { i = 1 } ^ { 4 } -2i^2 + \sum _ { i = 1 } ^ { 4 } i$. Using the formulas, we find $\sum _ { i = 1 } ^ { 4 } i = \frac{4(4+1)}{2} = 10$ and $\sum _ { i = 1 } ^ { 4 } i^2 = \frac{4(4+1)(2*4+1)}{6} = \frac{4*5*9}{6} = 30$. Thus, the series equals $-2*30 + 10 = -60 + 10 = -50$.

Use the Summation Properties to Evaluate the Series - ∑i=14(−2i2+i)\sum _ { i = 1 } ^ { 4 } \left( - 2 i ^ { 2 } + i \right)i=1∑4​(−2i2+i)

Use the Summation Properties to Evaluate the Series - $\sum _ { i = 1 } ^ { 4 } \left( - 2 i ^ { 2 } + i \right)$