Calculus Examples

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

Step 1

Combine the numerators over the common denominator.

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

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $n$ by $n$ .

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

Step 2.3

Multiply $n$ by $1$ .

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

Step 2.4

Expand $(n^{2} + n) (2 n + 1)$ using the FOIL Method.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Apply the distributive property.

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

Step 2.5

Simplify and combine like terms.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.5.1.2

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

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Step 2.5.1.2.1

Move $n$ .

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

Step 2.5.1.2.2

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

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Step 2.5.1.2.2.1

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

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

Step 2.5.1.2.2.2

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

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

Step 2.5.1.2.3

Add $1$ and $2$ .

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

Step 2.5.1.3

Multiply $n^{2}$ by $1$ .

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

Step 2.5.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.5.1.5

Multiply $n$ by $n$ by adding the exponents.

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Step 2.5.1.5.1

Move $n$ .

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

Step 2.5.1.5.2

Multiply $n$ by $n$ .

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

Step 2.5.1.6

Multiply $n$ by $1$ .

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

Step 2.5.2

Add $n^{2}$ and $2 n^{2}$ .

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

Step 2.6

Apply the distributive property.

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

Step 2.7

Multiply $n$ by $n$ by adding the exponents.

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Step 2.7.1

Move $n$ .

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

Step 2.7.2

Multiply $n$ by $n$ .

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

Step 2.8

Multiply $- 1$ by $1$ .

$\frac{2 n^{3} + 3 n^{2} + n - n^{2} - n}{6}$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $2 n^{3} + 3 n^{2} + n - n^{2} - n$ .

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Step 3.1.1

Subtract $n$ from $n$ .

$\frac{2 n^{3} + 3 n^{2} - n^{2} + 0}{6}$

Step 3.1.2

Add $2 n^{3} + 3 n^{2} - n^{2}$ and $0$ .

$\frac{2 n^{3} + 3 n^{2} - n^{2}}{6}$

Step 3.2

Subtract $n^{2}$ from $3 n^{2}$ .

$\frac{2 n^{3} + 2 n^{2}}{6}$

Step 3.3

Factor $2 n^{2}$ out of $2 n^{3} + 2 n^{2}$ .

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Step 3.3.1

Factor $2 n^{2}$ out of $2 n^{3}$ .

$\frac{2 n^{2} (n) + 2 n^{2}}{6}$

Step 3.3.2

Factor $2 n^{2}$ out of $2 n^{2}$ .

$\frac{2 n^{2} (n) + 2 n^{2} (1)}{6}$

Step 3.3.3

Factor $2 n^{2}$ out of $2 n^{2} (n) + 2 n^{2} (1)$ .

$\frac{2 n^{2} (n + 1)}{6}$

Step 3.4

Cancel the common factor of $2$ and $6$ .

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Step 3.4.1

Factor $2$ out of $2 n^{2} (n + 1)$ .

$\frac{2 (n^{2} (n + 1))}{6}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $6$ .

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

$\frac{n^{2} (n + 1)}{3}$