Precalculus Examples

$\sum_{n = 3}^{8} 15 - 4 n$

Step 1

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

$\sum_{n = 3}^{8} 15 - 4 n = \sum_{n = 1}^{8} 15 - 4 n - \sum_{n = 1}^{2} 15 - 4 n$

Step 2

Evaluate $\sum_{n = 1}^{8} 15 - 4 n$ .

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Step 2.1

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

$\sum_{n = 1}^{8} 15 - 4 n = \sum_{n = 1}^{8} 15 - 4 \sum_{n = 1}^{8} n$

Step 2.2

Evaluate $\sum_{n = 1}^{8} 15$ .

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Step 2.2.1

The formula for the summation of a constant is:

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

Step 2.2.2

Substitute the values into the formula.

$(15) (8)$

Step 2.2.3

Multiply $15$ by $8$ .

$120$

Step 2.3

Evaluate $4 \sum_{n = 1}^{8} n$ .

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Step 2.3.1

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

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

Step 2.3.2

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

$(- 4) (\frac{8 (8 + 1)}{2})$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the expression.

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Step 2.3.3.1.1

Add $8$ and $1$ .

$- 4 \frac{8 \cdot 9}{2}$

Step 2.3.3.1.2

Multiply $8$ by $9$ .

$- 4 (\frac{72}{2})$

Step 2.3.3.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.1

Factor $2$ out of $- 4$ .

$2 (- 2) \frac{72}{2}$

Step 2.3.3.2.2

Cancel the common factor.

$2 \cdot - 2 \frac{72}{2}$

Step 2.3.3.2.3

Rewrite the expression.

$- 2 \cdot 72$

Step 2.3.3.3

Multiply $- 2$ by $72$ .

$- 144$

Step 2.4

Add the results of the summations.

$120 - 144$

Step 2.5

Subtract $144$ from $120$ .

$- 24$

Step 3

Evaluate $\sum_{n = 1}^{2} 15 - 4 n$ .

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Step 3.1

Expand the series for each value of $n$ .

$15 - 4 \cdot 1 + 15 - 4 \cdot 2$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $- 4$ by $1$ .

$15 - 4 + 15 - 4 \cdot 2$

Step 3.2.2

Subtract $4$ from $15$ .

$11 + 15 - 4 \cdot 2$

Step 3.2.3

Multiply $- 4$ by $2$ .

$11 + 15 - 8$

Step 3.2.4

Subtract $8$ from $15$ .

$11 + 7$

Step 3.2.5

Add $11$ and $7$ .

$18$

Step 4

Replace the summations with the values found.

$- 24 - 18$

Step 5

Subtract $18$ from $- 24$ .

$- 42$