Calculus Examples

$\sum_{k = 2}^{5} - 4 k + 3 k^{2}$

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Multiply $- 4$ by $2$ .

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

Step 2.2

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

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

Step 2.3

Multiply $3$ by $4$ .

$- 8 + 12 - 4 \cdot 3 + 3 \cdot 3^{2} - 4 \cdot 4 + 3 \cdot 4^{2} - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.4

Add $- 8$ and $12$ .

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

Step 2.5

Multiply $- 4$ by $3$ .

$4 - 12 + 3 \cdot 3^{2} - 4 \cdot 4 + 3 \cdot 4^{2} - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.6

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

$4 - 12 + 3 \cdot 9 - 4 \cdot 4 + 3 \cdot 4^{2} - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.7

Multiply $3$ by $9$ .

$4 - 12 + 27 - 4 \cdot 4 + 3 \cdot 4^{2} - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.8

Add $- 12$ and $27$ .

$4 + 15 - 4 \cdot 4 + 3 \cdot 4^{2} - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.9

Add $4$ and $15$ .

$19 - 4 \cdot 4 + 3 \cdot 4^{2} - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.10

Multiply $- 4$ by $4$ .

$19 - 16 + 3 \cdot 4^{2} - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.11

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

$19 - 16 + 3 \cdot 16 - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.12

Multiply $3$ by $16$ .

$19 - 16 + 48 - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.13

Add $- 16$ and $48$ .

$19 + 32 - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.14

Add $19$ and $32$ .

$51 - 4 \cdot 5 + 3 \cdot 5^{2}$

Step 2.15

Multiply $- 4$ by $5$ .

$51 - 20 + 3 \cdot 5^{2}$

Step 2.16

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

$51 - 20 + 3 \cdot 25$

Step 2.17

Multiply $3$ by $25$ .

$51 - 20 + 75$

Step 2.18

Add $- 20$ and $75$ .

$51 + 55$

Step 2.19

Add $51$ and $55$ .

$106$