Calculus Examples

$\sum_{k = 5}^{8} {(- 1)}^{k} (4 k - 5)$

Step 1

Expand the series for each value of $k$ .

${(- 1)}^{5} (4 \cdot 5 - 5) + {(- 1)}^{6} (4 \cdot 6 - 5) + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2

Simplify.

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Step 2.1

Raise $- 1$ to the power of $5$ .

$- 1 (4 \cdot 5 - 5) + {(- 1)}^{6} (4 \cdot 6 - 5) + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.2

Multiply $4$ by $5$ .

$- 1 (20 - 5) + {(- 1)}^{6} (4 \cdot 6 - 5) + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.3

Subtract $5$ from $20$ .

$- 1 \cdot 15 + {(- 1)}^{6} (4 \cdot 6 - 5) + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.4

Multiply $- 1$ by $15$ .

$- 15 + {(- 1)}^{6} (4 \cdot 6 - 5) + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.5

Raise $- 1$ to the power of $6$ .

$- 15 + 1 (4 \cdot 6 - 5) + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.6

Multiply $4$ by $6$ .

$- 15 + 1 (24 - 5) + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.7

Subtract $5$ from $24$ .

$- 15 + 1 \cdot 19 + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.8

Multiply $1$ by $19$ .

$- 15 + 19 + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.9

Add $- 15$ and $19$ .

$4 + {(- 1)}^{7} (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.10

Raise $- 1$ to the power of $7$ .

$4 - 1 (4 \cdot 7 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.11

Multiply $4$ by $7$ .

$4 - 1 (28 - 5) + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.12

Subtract $5$ from $28$ .

$4 - 1 \cdot 23 + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.13

Multiply $- 1$ by $23$ .

$4 - 23 + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.14

Subtract $23$ from $4$ .

$- 19 + {(- 1)}^{8} (4 \cdot 8 - 5)$

Step 2.15

Raise $- 1$ to the power of $8$ .

$- 19 + 1 (4 \cdot 8 - 5)$

Step 2.16

Multiply $4$ by $8$ .

$- 19 + 1 (32 - 5)$

Step 2.17

Subtract $5$ from $32$ .

$- 19 + 1 \cdot 27$

Step 2.18

Multiply $1$ by $27$ .

$- 19 + 27$

Step 2.19

Add $- 19$ and $27$ .

$8$