Calculus Examples

$\sum_{k = 1}^{n} 12 k + 8$

Step 1

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

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

Step 2

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

$(12) (\frac{n (n + 1)}{2})$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Factor $2$ out of $12$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

$6 (n (n + 1))$

Step 3.2

Apply the distributive property.

$6 (n \cdot n + n \cdot 1)$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $n$ by $n$ .

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

Step 3.3.2

Multiply $n$ by $1$ .

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

Step 3.4

Apply the distributive property.

$6 n^{2} + 6 n$