Use the summation properties to evaluate the series. The following rules may be needed: $$\begin{array} { l l l }
\mathrm { n } & \mathrm { n } & \mathrm { n } \
\sum \mathrm { i } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) } { 2 } ; & \sum \mathrm { i } ^ { 2 } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) ( 2 \mathrm { n } + 1 ) } { 6 } ; & \sum \mathrm { i } ^ { 3 } = \frac { \mathrm { n } ^ { 2 } ( \mathrm { n } + 1 ) ^ { 2 } } { 4 } \
\mathrm { i } = 1 & \mathrm { i } = 1 & \mathrm { i } = 1
\end{array}$$
-$$\sum _ { i = 1 } ^ { 5 } \left( - 2 i ^ { 2 } + 2 i - 6 \right)$$
A) 50
B) $- 110$
C) $- 86$
D) $- 50$

Question

Use the summation properties to evaluate the series. The following rules may be needed: $$\begin{array} { l l l } 
\mathrm { n } & \mathrm { n } & \mathrm { n } \
\sum \mathrm { i } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) } { 2 } ; & \sum \mathrm { i } ^ { 2 } = \frac { \mathrm { n } ( \mathrm { n } + 1 ) ( 2 \mathrm { n } + 1 ) } { 6 } ; & \sum \mathrm { i } ^ { 3 } = \frac { \mathrm { n } ^ { 2 } ( \mathrm { n } + 1 ) ^ { 2 } } { 4 } \
\mathrm { i } = 1 & \mathrm { i } = 1 & \mathrm { i } = 1
\end{array}$$
-$$\sum _ { i = 1 } ^ { 5 } \left( - 2 i ^ { 2 } + 2 i - 6 \right)$$
A) 50
B) $- 110$
C) $- 86$
D) $- 50$

Quizplus · Accepted Answer

The correct answer is B because:$\sum _ { i = 1 } ^ { 5 } \left( - 2 i ^ { 2 } + 2 i - 6 \right) = \sum _ { i = 1 } ^ { 5 } - 2 i ^ { 2 } + \sum _ { i = 1 } ^ { 5 } 2 i - \sum _ { i = 1 } ^ { 5 } 6$$\Rightarrow - 2 \sum _ { i = 1 } ^ { 5 } i ^ { 2 } + 2 \sum _ { i = 1 } ^ { 5 } i - 6 \sum _ { i = 1 } ^ { 5 } 1$$\Rightarrow - 2 \left( \frac { 5 ( 5 + 1 ) ( 2 ( 5 ) + 1 ) } { 6 } \right) + 2 \left( \frac { 5 ( 5 + 1 ) } { 2 } \right) - 6 ( 5 )$$\Rightarrow - 2 \left( \frac { 5 ( 6 ) ( 11 ) } { 6 } \right) + 2 \left( \frac { 5 ( 6 ) } { 2 } \right) - 30$$\Rightarrow - 2 ( 55 ) + 30 - 30$$\Rightarrow - 110$

Use the Summation Properties to Evaluate the Series - ∑i=15(−2i2+2i−6)\sum _ { i = 1 } ^ { 5 } \left( - 2 i ^ { 2 } + 2 i - 6 \right)i=1∑5​(−2i2+2i−6)

Use the Summation Properties to Evaluate the Series - $\sum _ { i = 1 } ^ { 5 } \left( - 2 i ^ { 2 } + 2 i - 6 \right)$