Precalculus Examples

$\sum_{i = 1}^{22} 6 i - 5$

Step 1

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

$\sum_{k = 1}^{22} 6 i - 5 = 6 \sum_{k = 1}^{22} i + \sum_{k = 1}^{22} - 5$

Step 2

Evaluate $6 \sum_{k = 1}^{22} i$ .

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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.

$(6) (\frac{22 (22 + 1)}{2})$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the expression.

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Step 2.3.1.1

Add $22$ and $1$ .

$6 \frac{22 \cdot 23}{2}$

Step 2.3.1.2

Multiply $22$ by $23$ .

$6 (\frac{506}{2})$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Factor $2$ out of $6$ .

$2 (3) \frac{506}{2}$

Step 2.3.2.2

Cancel the common factor.

$2 \cdot 3 \frac{506}{2}$

Step 2.3.2.3

Rewrite the expression.

$3 \cdot 506$

Step 2.3.3

Multiply $3$ by $506$ .

$1518$

Step 3

Evaluate $\sum_{k = 1}^{22} - 5$ .

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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.

$(- 5) (22)$

Step 3.3

Multiply $- 5$ by $22$ .

$- 110$

Step 4

Add the results of the summations.

$1518 - 110$

Step 5

Subtract $110$ from $1518$ .

$1408$