Calculus Examples

$\sum_{n = 1}^{15} 5 (- 2 n)$

Step 1

Simplify the summation.

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Step 1.1

Multiply $- 2$ by $5$ .

$- 10 n$

Step 1.2

Rewrite the summation.

$\sum_{n = 1}^{15} - 10 n$

Step 2

Factor $- 10$ out of the summation.

$(- 10) \sum_{n = 1}^{15} n$

Step 3

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

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

Step 4

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

$(- 10) (\frac{15 (15 + 1)}{2})$

Step 5

Simplify.

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Step 5.1

Simplify the expression.

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Step 5.1.1

Add $15$ and $1$ .

$- 10 \frac{15 \cdot 16}{2}$

Step 5.1.2

Multiply $15$ by $16$ .

$- 10 (\frac{240}{2})$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $- 10$ .

$2 (- 5) \frac{240}{2}$

Step 5.2.2

Cancel the common factor.

$2 \cdot - 5 \frac{240}{2}$

Step 5.2.3

Rewrite the expression.

$- 5 \cdot 240$

Step 5.3

Multiply $- 5$ by $240$ .

$- 1200$