Calculus Examples

$\sum_{k = 1}^{8} - 4 k^{2} + 6 k$

Step 1

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

$\sum_{k = 1}^{8} - 4 k^{2} + 6 k = - 4 \sum_{k = 1}^{8} k^{2} + 6 \sum_{k = 1}^{8} k$

Step 2

Evaluate $4 \sum_{k = 1}^{8} k^{2}$ .

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Step 2.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 2.2

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

$(- 4) (\frac{8 (8 + 1) (2 \cdot 8 + 1)}{6})$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Add $8$ and $1$ .

$- 4 \frac{8 \cdot 9 (2 \cdot 8 + 1)}{6}$

Step 2.3.1.2

Combine exponents.

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Step 2.3.1.2.1

Multiply $8$ by $9$ .

$- 4 \frac{72 (2 \cdot 8 + 1)}{6}$

Step 2.3.1.2.2

Multiply $2$ by $8$ .

$- 4 \frac{72 (16 + 1)}{6}$

Step 2.3.1.3

Add $16$ and $1$ .

$- 4 \frac{72 \cdot 17}{6}$

Step 2.3.2

Simplify terms.

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Step 2.3.2.1

Multiply $72$ by $17$ .

$- 4 (\frac{1224}{6})$

Step 2.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.2.2.1

Factor $2$ out of $- 4$ .

$2 (- 2) \frac{1224}{6}$

Step 2.3.2.2.2

Factor $2$ out of $6$ .

$2 \cdot - 2 \frac{1224}{2 \cdot 3}$

Step 2.3.2.2.3

Cancel the common factor.

$2 \cdot - 2 \frac{1224}{2 \cdot 3}$

Step 2.3.2.2.4

Rewrite the expression.

$- 2 (\frac{1224}{3})$

Step 2.3.2.3

Combine $- 2$ and $\frac{1224}{3}$ .

$\frac{- 2 \cdot 1224}{3}$

Step 2.3.2.4

Simplify the expression.

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Step 2.3.2.4.1

Multiply $- 2$ by $1224$ .

$\frac{- 2448}{3}$

Step 2.3.2.4.2

Divide $- 2448$ by $3$ .

$- 816$

Step 3

Evaluate $6 \sum_{k = 1}^{8} k$ .

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Step 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 3.2

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

$(6) (\frac{8 (8 + 1)}{2})$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the expression.

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Step 3.3.1.1

Add $8$ and $1$ .

$6 \frac{8 \cdot 9}{2}$

Step 3.3.1.2

Multiply $8$ by $9$ .

$6 (\frac{72}{2})$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Factor $2$ out of $6$ .

$2 (3) \frac{72}{2}$

Step 3.3.2.2

Cancel the common factor.

$2 \cdot 3 \frac{72}{2}$

Step 3.3.2.3

Rewrite the expression.

$3 \cdot 72$

Step 3.3.3

Multiply $3$ by $72$ .

$216$

Step 4

Add the results of the summations.

$- 816 + 216$

Step 5

Add $- 816$ and $216$ .

$- 600$