Calculus Examples

$\sum_{i = 1}^{19} 3 i - 4$

Step 1

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

$\sum_{k = 1}^{19} 3 i - 4 = 3 \sum_{k = 1}^{19} i + \sum_{k = 1}^{19} - 4$

Step 2

Evaluate $3 \sum_{k = 1}^{19} i$ .

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Step 2.1

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

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

Step 2.2

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

$(3) (\frac{19 (19 + 1)}{2})$

Step 2.3

Simplify.

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Step 2.3.1

Add $19$ and $1$ .

$3 \frac{19 \cdot 20}{2}$

Step 2.3.2

Multiply $19$ by $20$ .

$3 (\frac{380}{2})$

Step 2.3.3

Divide $380$ by $2$ .

$3 \cdot 190$

Step 2.3.4

Multiply $3$ by $190$ .

$570$

Step 3

Evaluate $\sum_{k = 1}^{19} - 4$ .

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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.

$(- 4) (19)$

Step 3.3

Multiply $- 4$ by $19$ .

$- 76$

Step 4

Add the results of the summations.

$570 - 76$

Step 5

Subtract $76$ from $570$ .

$494$