Calculus Examples

$\sum_{i = 1}^{22} i {(i - 1)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(i - 1)}^{2}$ as $(i - 1) (i - 1)$ .

$i ((i - 1) (i - 1))$

Step 1.2

Expand $(i - 1) (i - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$i (i (i - 1) - 1 (i - 1))$

Step 1.2.2

Apply the distributive property.

$i (i \cdot i + i \cdot - 1 - 1 (i - 1))$

Step 1.2.3

Apply the distributive property.

$i (i \cdot i + i \cdot - 1 - 1 i - 1 \cdot - 1)$

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

Multiply $i$ by $i$ .

$i (i^{2} + i \cdot - 1 - 1 i - 1 \cdot - 1)$

Step 1.3.1.2

Move $- 1$ to the left of $i$ .

$i (i^{2} - 1 \cdot i - 1 i - 1 \cdot - 1)$

Step 1.3.1.3

Rewrite $- 1 i$ as $- i$ .

$i (i^{2} - i - 1 i - 1 \cdot - 1)$

Step 1.3.1.4

Rewrite $- 1 i$ as $- i$ .

$i (i^{2} - i - i - 1 \cdot - 1)$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

$i (i^{2} - i - i + 1)$

Step 1.3.2

Subtract $i$ from $- i$ .

$i (i^{2} - 2 i + 1)$

Step 1.4

Apply the distributive property.

$i \cdot i^{2} + i (- 2 i) + i \cdot 1$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $i$ by $i^{2}$ by adding the exponents.

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Step 1.5.1.1

Multiply $i$ by $i^{2}$ .

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Step 1.5.1.1.1

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

$i^{1} i^{2} + i (- 2 i) + i \cdot 1$

Step 1.5.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$i^{1 + 2} + i (- 2 i) + i \cdot 1$

Step 1.5.1.2

Add $1$ and $2$ .

$i^{3} + i (- 2 i) + i \cdot 1$

Step 1.5.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.3

Multiply $i$ by $1$ .

$i^{3} - 2 i \cdot i + i$

Step 1.6

Multiply $i$ by $i$ by adding the exponents.

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Step 1.6.1

Move $i$ .

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

Step 1.6.2

Multiply $i$ by $i$ .

$i^{3} - 2 i^{2} + i$

Step 1.7

Rewrite the summation.

$\sum_{i = 1}^{22} i^{3} - 2 i^{2} + i$

Step 2

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

$\sum_{i = 1}^{22} i^{3} - 2 i^{2} + i = \sum_{i = 1}^{22} i^{3} - 2 \sum_{i = 1}^{22} i^{2} + \sum_{i = 1}^{22} i$

Step 3

Evaluate $\sum_{i = 1}^{22} i^{3}$ .

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Step 3.1

The formula for the summation of a polynomial with degree $3$ is:

$\sum_{k = 1}^{n} i^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$

Step 3.2

Substitute the values into the formula.

$\frac{22^{2} {(22 + 1)}^{2}}{4}$

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 $22$ and $1$ .

$\frac{22^{2} \cdot 23^{2}}{4}$

Step 3.3.1.2

Raise $22$ to the power of $2$ .

$\frac{484 \cdot 23^{2}}{4}$

Step 3.3.1.3

Raise $23$ to the power of $2$ .

$\frac{484 \cdot 529}{4}$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $484$ by $529$ .

$\frac{256036}{4}$

Step 3.3.2.2

Divide $256036$ by $4$ .

$64009$

Step 4

Evaluate $2 \sum_{i = 1}^{22} i^{2}$ .

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Step 4.1

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 4.2

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

$(- 2) (\frac{22 (22 + 1) (2 \cdot 22 + 1)}{6})$

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 $22$ and $1$ .

$- 2 \frac{22 \cdot 23 (2 \cdot 22 + 1)}{6}$

Step 4.3.1.2

Combine exponents.

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Step 4.3.1.2.1

Multiply $22$ by $23$ .

$- 2 \frac{506 (2 \cdot 22 + 1)}{6}$

Step 4.3.1.2.2

Multiply $2$ by $22$ .

$- 2 \frac{506 (44 + 1)}{6}$

Step 4.3.1.3

Add $44$ and $1$ .

$- 2 \frac{506 \cdot 45}{6}$

Step 4.3.2

Reduce the expression by cancelling the common factors.

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Step 4.3.2.1

Multiply $506$ by $45$ .

$- 2 (\frac{22770}{6})$

Step 4.3.2.2

Cancel the common factor of $2$ .

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Step 4.3.2.2.1

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{22770}{6}$

Step 4.3.2.2.2

Factor $2$ out of $6$ .

$2 \cdot - 1 \frac{22770}{2 \cdot 3}$

Step 4.3.2.2.3

Cancel the common factor.

$2 \cdot - 1 \frac{22770}{2 \cdot 3}$

Step 4.3.2.2.4

Rewrite the expression.

$- 1 (\frac{22770}{3})$

Step 4.3.2.3

Simplify the expression.

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Step 4.3.2.3.1

Divide $22770$ by $3$ .

$- 1 \cdot 7590$

Step 4.3.2.3.2

Multiply $- 1$ by $7590$ .

$- 7590$

Step 5

Evaluate $\sum_{i = 1}^{22} i$ .

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Step 5.1

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

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

Step 5.2

Substitute the values into the formula.

$\frac{22 (22 + 1)}{2}$

Step 5.3

Simplify.

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Step 5.3.1

Cancel the common factor of $22$ and $2$ .

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Step 5.3.1.1

Factor $2$ out of $22 (22 + 1)$ .

$\frac{2 (11 (22 + 1))}{2}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (11 (22 + 1))}{2 (1)}$

Step 5.3.1.2.2

Cancel the common factor.

$\frac{2 (11 (22 + 1))}{2 \cdot 1}$

Step 5.3.1.2.3

Rewrite the expression.

$\frac{11 (22 + 1)}{1}$

Step 5.3.1.2.4

Divide $11 (22 + 1)$ by $1$ .

$11 (22 + 1)$

Step 5.3.2

Simplify the expression.

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Step 5.3.2.1

Add $22$ and $1$ .

$11 \cdot 23$

Step 5.3.2.2

Multiply $11$ by $23$ .

$253$

Step 6

Add the results of the summations.

$64009 - 7590 + 253$

Step 7

Simplify.

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Step 7.1

Subtract $7590$ from $64009$ .

$56419 + 253$

Step 7.2

Add $56419$ and $253$ .

$56672$