Calculus Examples

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

Step 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 2

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

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

Step 3

Simplify.

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Step 3.1

Simplify terms.

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Step 3.1.1

Cancel the common factor of $6$ .

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

$n (n + 1) (2 n + 1)$

Step 3.1.2

Apply the distributive property.

$(n \cdot n + n \cdot 1) (2 n + 1)$

Step 3.1.3

Simplify the expression.

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Step 3.1.3.1

Multiply $n$ by $n$ .

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

Step 3.1.3.2

Multiply $n$ by $1$ .

$(n^{2} + n) (2 n + 1)$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$n^{2} (2 n + 1) + n (2 n + 1)$

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $n$ .

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

Step 3.3.1.2.2

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

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Step 3.3.1.2.2.1

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

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

Step 3.3.1.2.2.2

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

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

Step 3.3.1.2.3

Add $1$ and $2$ .

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

Step 3.3.1.3

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

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

Step 3.3.1.4

Rewrite using the commutative property of multiplication.

$2 n^{3} + n^{2} + 2 n \cdot n + n \cdot 1$

Step 3.3.1.5

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

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Step 3.3.1.5.1

Move $n$ .

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

Step 3.3.1.5.2

Multiply $n$ by $n$ .

$2 n^{3} + n^{2} + 2 n^{2} + n \cdot 1$

Step 3.3.1.6

Multiply $n$ by $1$ .

$2 n^{3} + n^{2} + 2 n^{2} + n$

Step 3.3.2

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

$2 n^{3} + 3 n^{2} + n$