Calculus Examples

$\sum_{i = 4}^{10} \frac{i - i^{2}}{3}$

Step 1

Simplify the summation.

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Step 1.1

Factor $i$ out of $i - i^{2}$ .

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Step 1.1.1

Raise $i$ to the power of $1$ .

$\frac{i^{1} - i^{2}}{3}$

Step 1.1.2

Factor $i$ out of $i^{1}$ .

$\frac{i \cdot 1 - i^{2}}{3}$

Step 1.1.3

Factor $i$ out of $- i^{2}$ .

$\frac{i \cdot 1 + i (- i)}{3}$

Step 1.1.4

Factor $i$ out of $i \cdot 1 + i (- i)$ .

$\frac{i (1 - i)}{3}$

Step 1.2

Rewrite the summation.

$\sum_{i = 4}^{10} \frac{i (1 - i)}{3}$

Step 2

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

$\sum_{i = 4}^{10} \frac{i}{3} - \frac{i^{2}}{3} = \sum_{i = 1}^{10} \frac{i}{3} - \frac{i^{2}}{3} - \sum_{i = 1}^{3} \frac{i}{3} - \frac{i^{2}}{3}$

Step 3

Evaluate $\sum_{i = 1}^{10} \frac{i}{3} - \frac{i^{2}}{3}$ .

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Step 3.1

Simplify the summation.

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Step 3.1.1

Combine the numerators over the common denominator.

$\frac{i - i^{2}}{3}$

Step 3.1.2

Rewrite the summation.

$\sum_{i = 1}^{10} \frac{i - i^{2}}{3}$

Step 3.2

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

$\sum_{i = 1}^{10} \frac{i}{3} - \frac{i^{2}}{3} = \sum_{i = 1}^{10} \frac{i}{3} + \sum_{i = 1}^{10} - \frac{i^{2}}{3}$

Step 3.3

Evaluate $\sum_{i = 1}^{10} \frac{i}{3}$ .

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Step 3.3.1

Factor $\frac{1}{3}$ out of the summation.

$\frac{1}{3} \sum_{k = 1}^{10} i$

Step 3.3.2

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

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

Step 3.3.3

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

$(\frac{1}{3}) (\frac{10 (10 + 1)}{2})$

Step 3.3.4

Simplify.

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Step 3.3.4.1

Simplify the expression.

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Step 3.3.4.1.1

Add $10$ and $1$ .

$\frac{1}{3} \cdot \frac{10 \cdot 11}{2}$

Step 3.3.4.1.2

Multiply $10$ by $11$ .

$\frac{1}{3} \cdot \frac{110}{2}$

Step 3.3.4.1.3

Divide $110$ by $2$ .

$\frac{1}{3} \cdot 55$

Step 3.3.4.2

Combine $\frac{1}{3}$ and $55$ .

$\frac{55}{3}$

Step 3.4

Evaluate $\sum_{i = 1}^{10} - \frac{i^{2}}{3}$ .

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Step 3.4.1

Factor $- \frac{1}{3}$ out of the summation.

$- \frac{1}{3} \sum_{k = 1}^{10} i^{2}$

Step 3.4.2

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

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

Step 3.4.3

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

$(- \frac{1}{3}) (\frac{10 (10 + 1) (2 \cdot 10 + 1)}{6})$

Step 3.4.4

Simplify.

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Step 3.4.4.1

Simplify the numerator.

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Step 3.4.4.1.1

Add $10$ and $1$ .

$- \frac{1}{3} \cdot \frac{10 \cdot 11 (2 \cdot 10 + 1)}{6}$

Step 3.4.4.1.2

Combine exponents.

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Step 3.4.4.1.2.1

Multiply $10$ by $11$ .

$- \frac{1}{3} \cdot \frac{110 (2 \cdot 10 + 1)}{6}$

Step 3.4.4.1.2.2

Multiply $2$ by $10$ .

$- \frac{1}{3} \cdot \frac{110 (20 + 1)}{6}$

Step 3.4.4.1.3

Add $20$ and $1$ .

$- \frac{1}{3} \cdot \frac{110 \cdot 21}{6}$

Step 3.4.4.2

Reduce the expression by cancelling the common factors.

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Step 3.4.4.2.1

Multiply $110$ by $21$ .

$- \frac{1}{3} \cdot \frac{2310}{6}$

Step 3.4.4.2.2

Cancel the common factor of $3$ .

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Step 3.4.4.2.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{- 1}{3} \cdot \frac{2310}{6}$

Step 3.4.4.2.2.2

Factor $3$ out of $2310$ .

$\frac{- 1}{3} \cdot \frac{3 (770)}{6}$

Step 3.4.4.2.2.3

Cancel the common factor.

$\frac{- 1}{3} \cdot \frac{3 \cdot 770}{6}$

Step 3.4.4.2.2.4

Rewrite the expression.

$- 1 (\frac{770}{6})$

Step 3.4.4.2.3

Cancel the common factor of $770$ and $6$ .

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Step 3.4.4.2.3.1

Factor $2$ out of $770$ .

$- 1 \frac{2 (385)}{6}$

Step 3.4.4.2.3.2

Cancel the common factors.

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Step 3.4.4.2.3.2.1

Factor $2$ out of $6$ .

$- 1 \frac{2 \cdot 385}{2 \cdot 3}$

Step 3.4.4.2.3.2.2

Cancel the common factor.

$- 1 \frac{2 \cdot 385}{2 \cdot 3}$

Step 3.4.4.2.3.2.3

Rewrite the expression.

$- 1 (\frac{385}{3})$

Step 3.4.4.2.4

Rewrite $- 1 (\frac{385}{3})$ as $- (\frac{385}{3})$ .

$- \frac{385}{3}$

Step 3.5

Add the results of the summations.

$\frac{55}{3} - \frac{385}{3}$

Step 3.6

Simplify.

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Step 3.6.1

Combine the numerators over the common denominator.

$\frac{55 - 385}{3}$

Step 3.6.2

Simplify the expression.

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Step 3.6.2.1

Subtract $385$ from $55$ .

$\frac{- 330}{3}$

Step 3.6.2.2

Divide $- 330$ by $3$ .

$- 110$

Step 4

Evaluate $\sum_{i = 1}^{3} \frac{i}{3} - \frac{i^{2}}{3}$ .

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Step 4.1

Simplify the summation.

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Step 4.1.1

Combine the numerators over the common denominator.

$\frac{i - i^{2}}{3}$

Step 4.1.2

Rewrite the summation.

$\sum_{i = 1}^{3} \frac{i - i^{2}}{3}$

Step 4.2

Expand the series for each value of $i$ .

$\frac{1 - 1^{2}}{3} + \frac{2 - 2^{2}}{3} + \frac{3 - 3^{2}}{3}$

Step 4.3

Simplify.

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Step 4.3.1

Combine the numerators over the common denominator.

$\frac{1 - 1^{2} + 2 - 2^{2} + 3 - 3^{2}}{3}$

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

One to any power is one.

$\frac{1 - 1 \cdot 1 + 2 - 2^{2} + 3 - 3^{2}}{3}$

Step 4.3.2.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1 + 2 - 2^{2} + 3 - 3^{2}}{3}$

Step 4.3.2.3

Raise $2$ to the power of $2$ .

$\frac{1 - 1 + 2 - 1 \cdot 4 + 3 - 3^{2}}{3}$

Step 4.3.2.4

Multiply $- 1$ by $4$ .

$\frac{1 - 1 + 2 - 4 + 3 - 3^{2}}{3}$

Step 4.3.2.5

Raise $3$ to the power of $2$ .

$\frac{1 - 1 + 2 - 4 + 3 - 1 \cdot 9}{3}$

Step 4.3.2.6

Multiply $- 1$ by $9$ .

$\frac{1 - 1 + 2 - 4 + 3 - 9}{3}$

Step 4.3.3

Subtract $1$ from $1$ .

$\frac{0 + 2 - 4 + 3 - 9}{3}$

Step 4.3.4

Add $0$ and $2$ .

$\frac{2 - 4 + 3 - 9}{3}$

Step 4.3.5

Subtract $4$ from $2$ .

$\frac{- 2 + 3 - 9}{3}$

Step 4.3.6

Add $- 2$ and $3$ .

$\frac{1 - 9}{3}$

Step 4.3.7

Subtract $9$ from $1$ .

$\frac{- 8}{3}$

Step 4.3.8

Move the negative in front of the fraction.

$- \frac{8}{3}$

Step 5

Replace the summations with the values found.

$- 110 + \frac{8}{3}$

Step 6

Simplify.

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Step 6.1

To write $- 110$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 110 \cdot \frac{3}{3} + \frac{8}{3}$

Step 6.2

Combine $- 110$ and $\frac{3}{3}$ .

$\frac{- 110 \cdot 3}{3} + \frac{8}{3}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{- 110 \cdot 3 + 8}{3}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $- 110$ by $3$ .

$\frac{- 330 + 8}{3}$

Step 6.4.2

Add $- 330$ and $8$ .

$\frac{- 322}{3}$

Step 6.5

Move the negative in front of the fraction.

$- \frac{322}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{322}{3}$

Decimal Form:

$- 107.\overline{3}$

Mixed Number Form:

$- 107 \frac{1}{3}$