Precalculus Examples

$\sum_{k = 1}^{200} {(2 k - 3)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(2 k - 3)}^{2}$ as $(2 k - 3) (2 k - 3)$ .

$(2 k - 3) (2 k - 3)$

Step 1.2

Expand $(2 k - 3) (2 k - 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$2 k (2 k - 3) - 3 (2 k - 3)$

Step 1.2.2

Apply the distributive property.

$2 k (2 k) + 2 k \cdot - 3 - 3 (2 k - 3)$

Step 1.2.3

Apply the distributive property.

$2 k (2 k) + 2 k \cdot - 3 - 3 (2 k) - 3 \cdot - 3$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 k \cdot k + 2 k \cdot - 3 - 3 (2 k) - 3 \cdot - 3$

Step 1.3.1.2

Multiply $k$ by $k$ by adding the exponents.

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Step 1.3.1.2.1

Move $k$ .

$2 \cdot 2 (k \cdot k) + 2 k \cdot - 3 - 3 (2 k) - 3 \cdot - 3$

Step 1.3.1.2.2

Multiply $k$ by $k$ .

$2 \cdot 2 k^{2} + 2 k \cdot - 3 - 3 (2 k) - 3 \cdot - 3$

Step 1.3.1.3

Multiply $2$ by $2$ .

$4 k^{2} + 2 k \cdot - 3 - 3 (2 k) - 3 \cdot - 3$

Step 1.3.1.4

Multiply $- 3$ by $2$ .

$4 k^{2} - 6 k - 3 (2 k) - 3 \cdot - 3$

Step 1.3.1.5

Multiply $2$ by $- 3$ .

$4 k^{2} - 6 k - 6 k - 3 \cdot - 3$

Step 1.3.1.6

Multiply $- 3$ by $- 3$ .

$4 k^{2} - 6 k - 6 k + 9$

Step 1.3.2

Subtract $6 k$ from $- 6 k$ .

$4 k^{2} - 12 k + 9$

Step 1.4

Rewrite the summation.

$\sum_{k = 1}^{200} 4 k^{2} - 12 k + 9$

Step 2

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

$\sum_{k = 1}^{200} 4 k^{2} - 12 k + 9 = 4 \sum_{k = 1}^{200} k^{2} - 12 \sum_{k = 1}^{200} k + \sum_{k = 1}^{200} 9$

Step 3

Evaluate $4 \sum_{k = 1}^{200} k^{2}$ .

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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 and make sure to multiply by the front term.

$(4) (\frac{200 (200 + 1) (2 \cdot 200 + 1)}{6})$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Add $200$ and $1$ .

$4 \frac{200 \cdot 201 (2 \cdot 200 + 1)}{6}$

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Multiply $200$ by $201$ .

$4 \frac{40200 (2 \cdot 200 + 1)}{6}$

Step 3.3.1.2.2

Multiply $2$ by $200$ .

$4 \frac{40200 (400 + 1)}{6}$

Step 3.3.1.3

Add $400$ and $1$ .

$4 \frac{40200 \cdot 401}{6}$

Step 3.3.2

Simplify terms.

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Step 3.3.2.1

Multiply $40200$ by $401$ .

$4 (\frac{16120200}{6})$

Step 3.3.2.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1

Factor $2$ out of $4$ .

$2 (2) \frac{16120200}{6}$

Step 3.3.2.2.2

Factor $2$ out of $6$ .

$2 \cdot 2 \frac{16120200}{2 \cdot 3}$

Step 3.3.2.2.3

Cancel the common factor.

$2 \cdot 2 \frac{16120200}{2 \cdot 3}$

Step 3.3.2.2.4

Rewrite the expression.

$2 (\frac{16120200}{3})$

Step 3.3.2.3

Combine $2$ and $\frac{16120200}{3}$ .

$\frac{2 \cdot 16120200}{3}$

Step 3.3.2.4

Simplify the expression.

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Step 3.3.2.4.1

Multiply $2$ by $16120200$ .

$\frac{32240400}{3}$

Step 3.3.2.4.2

Divide $32240400$ by $3$ .

$10746800$

Step 4

Evaluate $12 \sum_{k = 1}^{200} k$ .

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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.

$(- 12) (\frac{200 (200 + 1)}{2})$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the expression.

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Step 4.3.1.1

Add $200$ and $1$ .

$- 12 \frac{200 \cdot 201}{2}$

Step 4.3.1.2

Multiply $200$ by $201$ .

$- 12 (\frac{40200}{2})$

Step 4.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.1

Factor $2$ out of $- 12$ .

$2 (- 6) \frac{40200}{2}$

Step 4.3.2.2

Cancel the common factor.

$2 \cdot - 6 \frac{40200}{2}$

Step 4.3.2.3

Rewrite the expression.

$- 6 \cdot 40200$

Step 4.3.3

Multiply $- 6$ by $40200$ .

$- 241200$

Step 5

Evaluate $\sum_{k = 1}^{200} 9$ .

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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.

$(9) (200)$

Step 5.3

Multiply $9$ by $200$ .

$1800$

Step 6

Add the results of the summations.

$10746800 - 241200 + 1800$

Step 7

Simplify.

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Step 7.1

Subtract $241200$ from $10746800$ .

$10505600 + 1800$

Step 7.2

Add $10505600$ and $1800$ .

$10507400$