Precalculus Examples

$\sum_{n = 1}^{18} 10 + 3 n$

Step 1

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

$\sum_{n = 1}^{18} 10 + 3 n = \sum_{n = 1}^{18} 10 + 3 \sum_{n = 1}^{18} n$

Step 2

Evaluate $\sum_{n = 1}^{18} 10$ .

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Step 2.1

The formula for the summation of a constant is:

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

Step 2.2

Substitute the values into the formula.

$(10) (18)$

Step 2.3

Multiply $10$ by $18$ .

$180$

Step 3

Evaluate $3 \sum_{n = 1}^{18} n$ .

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Step 3.1

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

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

Step 3.2

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

$(3) (\frac{18 (18 + 1)}{2})$

Step 3.3

Simplify.

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Step 3.3.1

Add $18$ and $1$ .

$3 \frac{18 \cdot 19}{2}$

Step 3.3.2

Multiply $18$ by $19$ .

$3 (\frac{342}{2})$

Step 3.3.3

Divide $342$ by $2$ .

$3 \cdot 171$

Step 3.3.4

Multiply $3$ by $171$ .

$513$

Step 4

Add the results of the summations.

$180 + 513$

Step 5

Add $180$ and $513$ .

$693$