Calculus Examples

$\sum_{k = 0}^{6} - k^{2} - 9$

Step 1

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

$\sum_{k = 0}^{6} - k^{2} - 9 = \sum_{k = 1}^{6} - k^{2} - 9 + \sum_{k = 0}^{0} - k^{2} - 9$

Step 2

Evaluate $\sum_{k = 1}^{6} - k^{2} - 9$ .

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Step 2.1

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

$\sum_{k = 1}^{6} - k^{2} - 9 = - 1 \sum_{k = 1}^{6} k^{2} + \sum_{k = 1}^{6} - 9$

Step 2.2

Evaluate $1 \sum_{k = 1}^{6} k^{2}$ .

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Step 2.2.1

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 2.2.2

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

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

Simplify the numerator.

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Step 2.2.3.1.1

Add $6$ and $1$ .

$- 1 \frac{6 \cdot 7 (2 \cdot 6 + 1)}{6}$

Step 2.2.3.1.2

Combine exponents.

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Step 2.2.3.1.2.1

Multiply $6$ by $7$ .

$- 1 \frac{42 (2 \cdot 6 + 1)}{6}$

Step 2.2.3.1.2.2

Multiply $2$ by $6$ .

$- 1 \frac{42 (12 + 1)}{6}$

Step 2.2.3.1.3

Add $12$ and $1$ .

$- 1 \frac{42 \cdot 13}{6}$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Multiply $42$ by $13$ .

$- 1 (\frac{546}{6})$

Step 2.2.3.2.2

Divide $546$ by $6$ .

$- 1 \cdot 91$

Step 2.2.3.2.3

Multiply $- 1$ by $91$ .

$- 91$

Step 2.3

Evaluate $\sum_{k = 1}^{6} - 9$ .

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Step 2.3.1

The formula for the summation of a constant is:

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

Step 2.3.2

Substitute the values into the formula.

$(- 9) (6)$

Step 2.3.3

Multiply $- 9$ by $6$ .

$- 54$

Step 2.4

Add the results of the summations.

$- 91 - 54$

Step 2.5

Subtract $54$ from $- 91$ .

$- 145$

Step 3

Evaluate $\sum_{k = 0}^{0} - k^{2} - 9$ .

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Step 3.1

Expand the series for each value of $k$ .

$- 0^{2} - 9$

Step 3.2

Simplify.

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Step 3.2.1

Raise $0$ to the power of $2$ .

$- 0 - 9$

Step 3.2.2

Multiply $- 1$ by $0$ .

$0 - 9$

Step 3.2.3

Subtract $9$ from $0$ .

$- 9$

Step 4

Replace the summations with the values found.

$- 145 - 9$

Step 5

Subtract $9$ from $- 145$ .

$- 154$