Calculus Examples

$2 \sum_{k = 3}^{8} 9 - 2 k$

Step 1

Split the summation to make the starting value of $k$ equal to $1$ .

$2 \sum_{k = 3}^{8} 9 - 2 k = 2 (\sum_{k = 1}^{8} 9 - 2 k - \sum_{k = 1}^{2} 9 - 2 k)$

Step 2

Evaluate $\sum_{k = 1}^{8} 9 - 2 k$ .

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Step 2.1

Split the summation into smaller summations that fit the summation rules.

$\sum_{k = 1}^{8} 9 - 2 k = \sum_{k = 1}^{8} 9 - 2 \sum_{k = 1}^{8} k$

Step 2.2

Evaluate $\sum_{k = 1}^{8} 9$ .

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Step 2.2.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 2.2.2

Substitute the values into the formula.

$(9) (8)$

Step 2.2.3

Multiply $9$ by $8$ .

$72$

Step 2.3

Evaluate $2 \sum_{k = 1}^{8} k$ .

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Step 2.3.1

The formula for the summation of a polynomial with degree $1$ is:

$\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}$

Step 2.3.2

Substitute the values into the formula and make sure to multiply by the front term.

$(- 2) (\frac{8 (8 + 1)}{2})$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the expression.

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Step 2.3.3.1.1

Add $8$ and $1$ .

$- 2 \frac{8 \cdot 9}{2}$

Step 2.3.3.1.2

Multiply $8$ by $9$ .

$- 2 (\frac{72}{2})$

Step 2.3.3.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.1

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{72}{2}$

Step 2.3.3.2.2

Cancel the common factor.

$2 \cdot - 1 \frac{72}{2}$

Step 2.3.3.2.3

Rewrite the expression.

$- 1 \cdot 72$

Step 2.3.3.3

Multiply $- 1$ by $72$ .

$- 72$

Step 2.4

Add the results of the summations.

$72 - 72$

Step 2.5

Subtract $72$ from $72$ .

$0$

Step 3

Evaluate $\sum_{k = 1}^{2} 9 - 2 k$ .

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Step 3.1

Expand the series for each value of $k$ .

$9 - 2 \cdot 1 + 9 - 2 \cdot 2$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $- 2$ by $1$ .

$9 - 2 + 9 - 2 \cdot 2$

Step 3.2.2

Subtract $2$ from $9$ .

$7 + 9 - 2 \cdot 2$

Step 3.2.3

Multiply $- 2$ by $2$ .

$7 + 9 - 4$

Step 3.2.4

Subtract $4$ from $9$ .

$7 + 5$

Step 3.2.5

Add $7$ and $5$ .

$12$

Step 4

Replace the summations with the values found.

$2 (0 - 12)$

Step 5

Simplify.

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Step 5.1

Subtract $12$ from $0$ .

$2 \cdot - 12$

Step 5.2

Multiply $2$ by $- 12$ .

$- 24$