Calculus Examples

$\sum_{k = 6}^{9} k (k - 2)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$k \cdot k + k \cdot - 2$

Step 1.2

Multiply $k$ by $k$ .

$k^{2} + k \cdot - 2$

Step 1.3

Move $- 2$ to the left of $k$ .

$k^{2} - 2 k$

Step 1.4

Rewrite the summation.

$\sum_{k = 6}^{9} k^{2} - 2 k$

Step 2

Expand the series for each value of $k$ .

$6^{2} - 2 \cdot 6 + 7^{2} - 2 \cdot 7 + 8^{2} - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3

Simplify.

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Step 3.1

Raise $6$ to the power of $2$ .

$36 - 2 \cdot 6 + 7^{2} - 2 \cdot 7 + 8^{2} - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3.2

Multiply $- 2$ by $6$ .

$36 - 12 + 7^{2} - 2 \cdot 7 + 8^{2} - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3.3

Subtract $12$ from $36$ .

$24 + 7^{2} - 2 \cdot 7 + 8^{2} - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3.4

Raise $7$ to the power of $2$ .

$24 + 49 - 2 \cdot 7 + 8^{2} - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3.5

Multiply $- 2$ by $7$ .

$24 + 49 - 14 + 8^{2} - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3.6

Subtract $14$ from $49$ .

$24 + 35 + 8^{2} - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3.7

Add $24$ and $35$ .

$59 + 8^{2} - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3.8

Raise $8$ to the power of $2$ .

$59 + 64 - 2 \cdot 8 + 9^{2} - 2 \cdot 9$

Step 3.9

Multiply $- 2$ by $8$ .

$59 + 64 - 16 + 9^{2} - 2 \cdot 9$

Step 3.10

Subtract $16$ from $64$ .

$59 + 48 + 9^{2} - 2 \cdot 9$

Step 3.11

Add $59$ and $48$ .

$107 + 9^{2} - 2 \cdot 9$

Step 3.12

Raise $9$ to the power of $2$ .

$107 + 81 - 2 \cdot 9$

Step 3.13

Multiply $- 2$ by $9$ .

$107 + 81 - 18$

Step 3.14

Subtract $18$ from $81$ .

$107 + 63$

Step 3.15

Add $107$ and $63$ .

$170$