Calculus Examples

$\sum_{k = 1}^{6} {(k)}^{3} - {(k - 1)}^{3}$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Use the Binomial Theorem.

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

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Multiply $- 1$ by $3$ .

$k^{3} - (k^{3} - 3 k^{2} + 3 k {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.1.2.2

Raise $- 1$ to the power of $2$ .

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

Step 1.1.2.3

Multiply $3$ by $1$ .

$k^{3} - (k^{3} - 3 k^{2} + 3 k + {(- 1)}^{3})$

Step 1.1.2.4

Raise $- 1$ to the power of $3$ .

$k^{3} - (k^{3} - 3 k^{2} + 3 k - 1)$

Step 1.1.3

Apply the distributive property.

$k^{3} - k^{3} - (- 3 k^{2}) - (3 k) - - 1$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Multiply $- 3$ by $- 1$ .

$k^{3} - k^{3} + 3 k^{2} - (3 k) - - 1$

Step 1.1.4.2

Multiply $3$ by $- 1$ .

$k^{3} - k^{3} + 3 k^{2} - 3 k - - 1$

Step 1.1.4.3

Multiply $- 1$ by $- 1$ .

$k^{3} - k^{3} + 3 k^{2} - 3 k + 1$

Step 1.2

Subtract $k^{3}$ from $k^{3}$ .

$0 + 3 k^{2} - 3 k + 1$

Step 1.3

Add $0$ and $3 k^{2}$ .

$3 k^{2} - 3 k + 1$

Step 1.4

Rewrite the summation.

$\sum_{k = 1}^{6} 3 k^{2} - 3 k + 1$

Step 2

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

$\sum_{k = 1}^{6} 3 k^{2} - 3 k + 1 = 3 \sum_{k = 1}^{6} k^{2} - 3 \sum_{k = 1}^{6} k + \sum_{k = 1}^{6} 1$

Step 3

Evaluate $3 \sum_{k = 1}^{6} k^{2}$ .

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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.

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

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 $6$ and $1$ .

$3 \frac{6 \cdot 7 (2 \cdot 6 + 1)}{6}$

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Multiply $6$ by $7$ .

$3 \frac{42 (2 \cdot 6 + 1)}{6}$

Step 3.3.1.2.2

Multiply $2$ by $6$ .

$3 \frac{42 (12 + 1)}{6}$

Step 3.3.1.3

Add $12$ and $1$ .

$3 \frac{42 \cdot 13}{6}$

Step 3.3.2

Reduce the expression by cancelling the common factors.

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Step 3.3.2.1

Multiply $42$ by $13$ .

$3 (\frac{546}{6})$

Step 3.3.2.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1

Factor $3$ out of $6$ .

$3 \frac{546}{3 (2)}$

Step 3.3.2.2.2

Cancel the common factor.

$3 \frac{546}{3 \cdot 2}$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{546}{2}$

Step 3.3.2.3

Divide $546$ by $2$ .

$273$

Step 4

Evaluate $3 \sum_{k = 1}^{6} k$ .

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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.

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

Step 4.3

Simplify.

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Step 4.3.1

Add $6$ and $1$ .

$- 3 \frac{6 \cdot 7}{2}$

Step 4.3.2

Multiply $6$ by $7$ .

$- 3 (\frac{42}{2})$

Step 4.3.3

Divide $42$ by $2$ .

$- 3 \cdot 21$

Step 4.3.4

Multiply $- 3$ by $21$ .

$- 63$

Step 5

Evaluate $\sum_{k = 1}^{6} 1$ .

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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.

$(1) (6)$

Step 5.3

Multiply $6$ by $1$ .

$6$

Step 6

Add the results of the summations.

$273 - 63 + 6$

Step 7

Simplify.

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Step 7.1

Subtract $63$ from $273$ .

$210 + 6$

Step 7.2

Add $210$ and $6$ .

$216$