Precalculus Examples

$\sum_{4}^{18} 6 x - 1$

Step 1

Split the summation to make the starting value of $x$ equal to $1$ .

$\sum_{4}^{18} 6 x - 1 = \sum_{x = 1}^{18} 6 x - 1 - \sum_{x = 1}^{3} 6 x - 1$

Step 2

Evaluate $\sum_{x = 1}^{18} 6 x - 1$ .

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Step 2.1

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

$\sum_{x = 1}^{18} 6 x - 1 = 6 \sum_{x = 1}^{18} x + \sum_{x = 1}^{18} - 1$

Step 2.2

Evaluate $6 \sum_{x = 1}^{18} x$ .

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Step 2.2.1

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

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

Step 2.2.2

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

$(6) (\frac{18 (18 + 1)}{2})$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Simplify the expression.

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Step 2.2.3.1.1

Add $18$ and $1$ .

$6 \frac{18 \cdot 19}{2}$

Step 2.2.3.1.2

Multiply $18$ by $19$ .

$6 (\frac{342}{2})$

Step 2.2.3.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.1

Factor $2$ out of $6$ .

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

Step 2.2.3.2.2

Cancel the common factor.

$2 \cdot 3 \frac{342}{2}$

Step 2.2.3.2.3

Rewrite the expression.

$3 \cdot 342$

Step 2.2.3.3

Multiply $3$ by $342$ .

$1026$

Step 2.3

Evaluate $\sum_{x = 1}^{18} - 1$ .

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Step 2.3.1

The formula for the summation of a constant is:

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

Step 2.3.2

Substitute the values into the formula.

$(- 1) (18)$

Step 2.3.3

Multiply $- 1$ by $18$ .

$- 18$

Step 2.4

Add the results of the summations.

$1026 - 18$

Step 2.5

Subtract $18$ from $1026$ .

$1008$

Step 3

Evaluate $\sum_{x = 1}^{3} 6 x - 1$ .

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Step 3.1

Expand the series for each value of $x$ .

$6 \cdot 1 - 1 + 6 \cdot 2 - 1 + 6 \cdot 3 - 1$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $6$ by $1$ .

$6 - 1 + 6 \cdot 2 - 1 + 6 \cdot 3 - 1$

Step 3.2.2

Subtract $1$ from $6$ .

$5 + 6 \cdot 2 - 1 + 6 \cdot 3 - 1$

Step 3.2.3

Multiply $6$ by $2$ .

$5 + 12 - 1 + 6 \cdot 3 - 1$

Step 3.2.4

Subtract $1$ from $12$ .

$5 + 11 + 6 \cdot 3 - 1$

Step 3.2.5

Add $5$ and $11$ .

$16 + 6 \cdot 3 - 1$

Step 3.2.6

Multiply $6$ by $3$ .

$16 + 18 - 1$

Step 3.2.7

Subtract $1$ from $18$ .

$16 + 17$

Step 3.2.8

Add $16$ and $17$ .

$33$

Step 4

Replace the summations with the values found.

$1008 - 33$

Step 5

Subtract $33$ from $1008$ .

$975$