Calculus Examples

$\sum_{n = 1}^{15} n - 4$

Step 1

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

$\sum_{n = 1}^{15} n - 4 = \sum_{n = 1}^{15} n + \sum_{n = 1}^{15} - 4$

Step 2

Evaluate $\sum_{n = 1}^{15} n$ .

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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.

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

Step 2.3

Simplify.

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Step 2.3.1

Add $15$ and $1$ .

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

Step 2.3.2

Multiply $15$ by $16$ .

$\frac{240}{2}$

Step 2.3.3

Divide $240$ by $2$ .

$120$

Step 3

Evaluate $\sum_{n = 1}^{15} - 4$ .

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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.

$(- 4) (15)$

Step 3.3

Multiply $- 4$ by $15$ .

$- 60$

Step 4

Add the results of the summations.

$120 - 60$

Step 5

Subtract $60$ from $120$ .

$60$