Calculus Examples

$\sum_{n = 1}^{6} - n - 4$

Step 1

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

$\sum_{n = 1}^{6} - n - 4 = - 1 \sum_{n = 1}^{6} n + \sum_{n = 1}^{6} - 4$

Step 2

Evaluate $1 \sum_{n = 1}^{6} n$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify.

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Step 2.3.1

Add $6$ and $1$ .

$- 1 \frac{6 \cdot 7}{2}$

Step 2.3.2

Multiply $6$ by $7$ .

$- 1 (\frac{42}{2})$

Step 2.3.3

Divide $42$ by $2$ .

$- 1 \cdot 21$

Step 2.3.4

Multiply $- 1$ by $21$ .

$- 21$

Step 3

Evaluate $\sum_{n = 1}^{6} - 4$ .

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Step 3.1

The formula for the summation of a constant is:

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

Step 3.2

Substitute the values into the formula.

$(- 4) (6)$

Step 3.3

Multiply $- 4$ by $6$ .

$- 24$

Step 4

Add the results of the summations.

$- 21 - 24$

Step 5

Subtract $24$ from $- 21$ .

$- 45$