Precalculus Examples

$\sum_{k = 1}^{6} (k - 2) (k + 1)$

Step 1

Simplify the summation.

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Step 1.1

Expand $(k - 2) (k + 1)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$k (k + 1) - 2 (k + 1)$

Step 1.1.2

Apply the distributive property.

$k \cdot k + k \cdot 1 - 2 (k + 1)$

Step 1.1.3

Apply the distributive property.

$k \cdot k + k \cdot 1 - 2 k - 2 \cdot 1$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $k$ by $k$ .

$k^{2} + k \cdot 1 - 2 k - 2 \cdot 1$

Step 1.2.1.2

Multiply $k$ by $1$ .

$k^{2} + k - 2 k - 2 \cdot 1$

Step 1.2.1.3

Multiply $- 2$ by $1$ .

$k^{2} + k - 2 k - 2$

Step 1.2.2

Subtract $2 k$ from $k$ .

$k^{2} - k - 2$

Step 1.3

Rewrite the summation.

$\sum_{k = 1}^{6} k^{2} - k - 2$

Step 2

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

$\sum_{k = 1}^{6} k^{2} - k - 2 = \sum_{k = 1}^{6} k^{2} - 1 \sum_{k = 1}^{6} k + \sum_{k = 1}^{6} - 2$

Step 3

Evaluate $\sum_{k = 1}^{6} k^{2}$ .

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Step 3.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 3.2

Substitute the values into the formula.

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

Step 3.3

Simplify.

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Step 3.3.1

Cancel the common factor of $6$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Divide $(6 + 1) (2 \cdot 6 + 1)$ by $1$ .

$(6 + 1) (2 \cdot 6 + 1)$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Add $6$ and $1$ .

$7 (2 \cdot 6 + 1)$

Step 3.3.2.2

Multiply $2$ by $6$ .

$7 (12 + 1)$

Step 3.3.2.3

Add $12$ and $1$ .

$7 \cdot 13$

Step 3.3.2.4

Multiply $7$ by $13$ .

$91$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify.

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Step 4.3.1

Add $6$ and $1$ .

$- 1 \frac{6 \cdot 7}{2}$

Step 4.3.2

Multiply $6$ by $7$ .

$- 1 (\frac{42}{2})$

Step 4.3.3

Divide $42$ by $2$ .

$- 1 \cdot 21$

Step 4.3.4

Multiply $- 1$ by $21$ .

$- 21$

Step 5

Evaluate $\sum_{k = 1}^{6} - 2$ .

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Step 5.1

The formula for the summation of a constant is:

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

Step 5.2

Substitute the values into the formula.

$(- 2) (6)$

Step 5.3

Multiply $- 2$ by $6$ .

$- 12$

Step 6

Add the results of the summations.

$91 - 21 - 12$

Step 7

Simplify.

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Step 7.1

Subtract $21$ from $91$ .

$70 - 12$

Step 7.2

Subtract $12$ from $70$ .

$58$