Calculus Examples

$\sum_{k = 5}^{5} {(k + 1)}^{2} (k - 3)$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(k + 1)}^{2}$ as $(k + 1) (k + 1)$ .

$(k + 1) (k + 1) (k - 3)$

Step 1.2

Expand $(k + 1) (k + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$(k (k + 1) + 1 (k + 1)) (k - 3)$

Step 1.2.2

Apply the distributive property.

$(k \cdot k + k \cdot 1 + 1 (k + 1)) (k - 3)$

Step 1.2.3

Apply the distributive property.

$(k \cdot k + k \cdot 1 + 1 k + 1 \cdot 1) (k - 3)$

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

Multiply $k$ by $k$ .

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

Step 1.3.1.2

Multiply $k$ by $1$ .

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

Step 1.3.1.3

Multiply $k$ by $1$ .

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

Step 1.3.1.4

Multiply $1$ by $1$ .

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

Step 1.3.2

Add $k$ and $k$ .

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

Step 1.4

Expand $(k^{2} + 2 k + 1) (k - 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.5

Simplify each term.

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Step 1.5.1

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

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Step 1.5.1.1

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

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Step 1.5.1.1.1

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

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

Step 1.5.1.1.2

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

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

Step 1.5.1.2

Add $2$ and $1$ .

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

Step 1.5.2

Move $- 3$ to the left of $k^{2}$ .

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

Step 1.5.3

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

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Step 1.5.3.1

Move $k$ .

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

Step 1.5.3.2

Multiply $k$ by $k$ .

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

Step 1.5.4

Multiply $- 3$ by $2$ .

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

Step 1.5.5

Multiply $k$ by $1$ .

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

Step 1.5.6

Multiply $- 3$ by $1$ .

$k^{3} - 3 k^{2} + 2 k^{2} - 6 k + k - 3$

Step 1.6

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

$k^{3} - k^{2} - 6 k + k - 3$

Step 1.7

Add $- 6 k$ and $k$ .

$k^{3} - k^{2} - 5 k - 3$

Step 1.8

Rewrite the summation.

$\sum_{k = 5}^{5} k^{3} - k^{2} - 5 k - 3$

Step 2

Expand the series for each value of $k$ .

$5^{3} - 5^{2} - 5 \cdot 5 - 3$

Step 3

Simplify.

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Step 3.1

Raise $5$ to the power of $3$ .

$125 - 5^{2} - 5 \cdot 5 - 3$

Step 3.2

Raise $5$ to the power of $2$ .

$125 - 1 \cdot 25 - 5 \cdot 5 - 3$

Step 3.3

Multiply $- 1$ by $25$ .

$125 - 25 - 5 \cdot 5 - 3$

Step 3.4

Subtract $25$ from $125$ .

$100 - 5 \cdot 5 - 3$

Step 3.5

Multiply $- 5$ by $5$ .

$100 - 25 - 3$

Step 3.6

Subtract $25$ from $100$ .

$75 - 3$

Step 3.7

Subtract $3$ from $75$ .

$72$