Calculus Examples

$3 \sum_{t = 2}^{5} 2 t^{2} - 8 t$

Step 1

Expand the series for each value of $t$ .

$3 (2 \cdot 2^{2} - 8 \cdot 2 + 2 \cdot 3^{2} - 8 \cdot 3 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2

Simplify.

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Step 2.1

Raise $2$ to the power of $2$ .

$3 (2 \cdot 4 - 8 \cdot 2 + 2 \cdot 3^{2} - 8 \cdot 3 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.2

Multiply $2$ by $4$ .

$3 (8 - 8 \cdot 2 + 2 \cdot 3^{2} - 8 \cdot 3 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.3

Multiply $- 8$ by $2$ .

$3 (8 - 16 + 2 \cdot 3^{2} - 8 \cdot 3 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.4

Subtract $16$ from $8$ .

$3 (- 8 + 2 \cdot 3^{2} - 8 \cdot 3 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.5

Raise $3$ to the power of $2$ .

$3 (- 8 + 2 \cdot 9 - 8 \cdot 3 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.6

Multiply $2$ by $9$ .

$3 (- 8 + 18 - 8 \cdot 3 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.7

Multiply $- 8$ by $3$ .

$3 (- 8 + 18 - 24 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.8

Subtract $24$ from $18$ .

$3 (- 8 - 6 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.9

Subtract $6$ from $- 8$ .

$3 (- 14 + 2 \cdot 4^{2} - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.10

Raise $4$ to the power of $2$ .

$3 (- 14 + 2 \cdot 16 - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.11

Multiply $2$ by $16$ .

$3 (- 14 + 32 - 8 \cdot 4 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.12

Multiply $- 8$ by $4$ .

$3 (- 14 + 32 - 32 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.13

Subtract $32$ from $32$ .

$3 (- 14 + 0 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.14

Add $- 14$ and $0$ .

$3 (- 14 + 2 \cdot 5^{2} - 8 \cdot 5)$

Step 2.15

Raise $5$ to the power of $2$ .

$3 (- 14 + 2 \cdot 25 - 8 \cdot 5)$

Step 2.16

Multiply $2$ by $25$ .

$3 (- 14 + 50 - 8 \cdot 5)$

Step 2.17

Multiply $- 8$ by $5$ .

$3 (- 14 + 50 - 40)$

Step 2.18

Subtract $40$ from $50$ .

$3 (- 14 + 10)$

Step 2.19

Add $- 14$ and $10$ .

$3 \cdot - 4$

Step 2.20

Multiply $3$ by $- 4$ .

$- 12$