Calculus Examples

$2 \sum_{k = 1}^{10} 100 - {(- 10 + 2 k)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(- 10 + 2 k)}^{2}$ as $(- 10 + 2 k) (- 10 + 2 k)$ .

$100 - ((- 10 + 2 k) (- 10 + 2 k))$

Step 1.1.2

Expand $(- 10 + 2 k) (- 10 + 2 k)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$100 - (- 10 (- 10 + 2 k) + 2 k (- 10 + 2 k))$

Step 1.1.2.2

Apply the distributive property.

$100 - (- 10 \cdot - 10 - 10 (2 k) + 2 k (- 10 + 2 k))$

Step 1.1.2.3

Apply the distributive property.

$100 - (- 10 \cdot - 10 - 10 (2 k) + 2 k \cdot - 10 + 2 k (2 k))$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $- 10$ by $- 10$ .

$100 - (100 - 10 (2 k) + 2 k \cdot - 10 + 2 k (2 k))$

Step 1.1.3.1.2

Multiply $2$ by $- 10$ .

$100 - (100 - 20 k + 2 k \cdot - 10 + 2 k (2 k))$

Step 1.1.3.1.3

Multiply $- 10$ by $2$ .

$100 - (100 - 20 k - 20 k + 2 k (2 k))$

Step 1.1.3.1.4

Rewrite using the commutative property of multiplication.

$100 - (100 - 20 k - 20 k + 2 \cdot 2 k \cdot k)$

Step 1.1.3.1.5

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

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Step 1.1.3.1.5.1

Move $k$ .

$100 - (100 - 20 k - 20 k + 2 \cdot 2 (k \cdot k))$

Step 1.1.3.1.5.2

Multiply $k$ by $k$ .

$100 - (100 - 20 k - 20 k + 2 \cdot 2 k^{2})$

Step 1.1.3.1.6

Multiply $2$ by $2$ .

$100 - (100 - 20 k - 20 k + 4 k^{2})$

Step 1.1.3.2

Subtract $20 k$ from $- 20 k$ .

$100 - (100 - 40 k + 4 k^{2})$

Step 1.1.4

Apply the distributive property.

$100 - 1 \cdot 100 - (- 40 k) - (4 k^{2})$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply $- 1$ by $100$ .

$100 - 100 - (- 40 k) - (4 k^{2})$

Step 1.1.5.2

Multiply $- 40$ by $- 1$ .

$100 - 100 + 40 k - (4 k^{2})$

Step 1.1.5.3

Multiply $4$ by $- 1$ .

$100 - 100 + 40 k - 4 k^{2}$

Step 1.2

Subtract $100$ from $100$ .

$0 + 40 k - 4 k^{2}$

Step 1.3

Add $0$ and $40 k$ .

$40 k - 4 k^{2}$

Step 1.4

Rewrite the summation.

$2 \sum_{k = 1}^{10} 40 k - 4 k^{2}$

Step 2

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

$2 \sum_{k = 1}^{10} 40 k - 4 k^{2} = (2 \cdot 40) \sum_{k = 1}^{10} k + (2 \cdot - 4) \sum_{k = 1}^{10} k^{2}$

Step 3

Evaluate $(2 \cdot 40) \sum_{k = 1}^{10} k$ .

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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.

$(2 \cdot 40) (\frac{10 (10 + 1)}{2})$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the expression.

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Step 3.3.1.1

Add $10$ and $1$ .

$2 \cdot 40 \frac{10 \cdot 11}{2}$

Step 3.3.1.2

Multiply $10$ by $11$ .

$2 \cdot 40 \frac{110}{2}$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Factor $2$ out of $2 \cdot 40$ .

$2 (40) \frac{110}{2}$

Step 3.3.2.2

Cancel the common factor.

$2 \cdot 40 \frac{110}{2}$

Step 3.3.2.3

Rewrite the expression.

$40 \cdot 110$

Step 3.3.3

Multiply $40$ by $110$ .

$4400$

Step 4

Evaluate $(2 \cdot - 4) \sum_{k = 1}^{10} k^{2}$ .

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Step 4.1

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 4.2

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

$(2 \cdot - 4) (\frac{10 (10 + 1) (2 \cdot 10 + 1)}{6})$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Add $10$ and $1$ .

$2 \cdot - 4 \frac{10 \cdot 11 (2 \cdot 10 + 1)}{6}$

Step 4.3.1.2

Combine exponents.

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Step 4.3.1.2.1

Multiply $10$ by $11$ .

$2 \cdot - 4 \frac{110 (2 \cdot 10 + 1)}{6}$

Step 4.3.1.2.2

Multiply $2$ by $10$ .

$2 \cdot - 4 \frac{110 (20 + 1)}{6}$

Step 4.3.1.3

Add $20$ and $1$ .

$2 \cdot - 4 \frac{110 \cdot 21}{6}$

Step 4.3.2

Simplify terms.

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Step 4.3.2.1

Multiply $110$ by $21$ .

$2 \cdot - 4 \frac{2310}{6}$

Step 4.3.2.2

Multiply $2$ by $- 4$ .

$- 8 (\frac{2310}{6})$

Step 4.3.2.3

Cancel the common factor of $2$ .

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Step 4.3.2.3.1

Factor $2$ out of $- 8$ .

$2 (- 4) \frac{2310}{6}$

Step 4.3.2.3.2

Factor $2$ out of $6$ .

$2 \cdot - 4 \frac{2310}{2 \cdot 3}$

Step 4.3.2.3.3

Cancel the common factor.

$2 \cdot - 4 \frac{2310}{2 \cdot 3}$

Step 4.3.2.3.4

Rewrite the expression.

$- 4 (\frac{2310}{3})$

Step 4.3.2.4

Combine $- 4$ and $\frac{2310}{3}$ .

$\frac{- 4 \cdot 2310}{3}$

Step 4.3.2.5

Simplify the expression.

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Step 4.3.2.5.1

Multiply $- 4$ by $2310$ .

$\frac{- 9240}{3}$

Step 4.3.2.5.2

Divide $- 9240$ by $3$ .

$- 3080$

Step 5

Add the results of the summations.

$4400 - 3080$

Step 6

Subtract $3080$ from $4400$ .

$1320$