Precalculus Examples

$\sum_{k = 1}^{24} k^{2} - 7 k + 2$

Step 1

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

$\sum_{k = 1}^{24} k^{2} - 7 k + 2 = \sum_{k = 1}^{24} k^{2} - 7 \sum_{k = 1}^{24} k + \sum_{k = 1}^{24} 2$

Step 2

Evaluate $\sum_{k = 1}^{24} k^{2}$ .

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Step 2.1

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 2.2

Substitute the values into the formula.

$\frac{24 (24 + 1) (2 \cdot 24 + 1)}{6}$

Step 2.3

Simplify.

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Step 2.3.1

Cancel the common factor of $24$ and $6$ .

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Step 2.3.1.1

Factor $6$ out of $24 (24 + 1) (2 \cdot 24 + 1)$ .

$\frac{6 (4 (24 + 1) (2 \cdot 24 + 1))}{6}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $6$ out of $6$ .

$\frac{6 (4 (24 + 1) (2 \cdot 24 + 1))}{6 (1)}$

Step 2.3.1.2.2

Cancel the common factor.

$\frac{6 (4 (24 + 1) (2 \cdot 24 + 1))}{6 \cdot 1}$

Step 2.3.1.2.3

Rewrite the expression.

$\frac{4 (24 + 1) (2 \cdot 24 + 1)}{1}$

Step 2.3.1.2.4

Divide $4 (24 + 1) (2 \cdot 24 + 1)$ by $1$ .

$4 (24 + 1) (2 \cdot 24 + 1)$

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Add $24$ and $1$ .

$4 \cdot 25 (2 \cdot 24 + 1)$

Step 2.3.2.2

Multiply $4$ by $25$ .

$100 (2 \cdot 24 + 1)$

Step 2.3.2.3

Multiply $2$ by $24$ .

$100 (48 + 1)$

Step 2.3.2.4

Add $48$ and $1$ .

$100 \cdot 49$

Step 2.3.2.5

Multiply $100$ by $49$ .

$4900$

Step 3

Evaluate $7 \sum_{k = 1}^{24} k$ .

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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.

$(- 7) (\frac{24 (24 + 1)}{2})$

Step 3.3

Simplify.

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Step 3.3.1

Add $24$ and $1$ .

$- 7 \frac{24 \cdot 25}{2}$

Step 3.3.2

Multiply $24$ by $25$ .

$- 7 (\frac{600}{2})$

Step 3.3.3

Divide $600$ by $2$ .

$- 7 \cdot 300$

Step 3.3.4

Multiply $- 7$ by $300$ .

$- 2100$

Step 4

Evaluate $\sum_{k = 1}^{24} 2$ .

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Step 4.1

The formula for the summation of a constant is:

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

Step 4.2

Substitute the values into the formula.

$(2) (24)$

Step 4.3

Multiply $2$ by $24$ .

$48$

Step 5

Add the results of the summations.

$4900 - 2100 + 48$

Step 6

Simplify.

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Step 6.1

Subtract $2100$ from $4900$ .

$2800 + 48$

Step 6.2

Add $2800$ and $48$ .

$2848$