Calculus Examples

$\sum_{n = 1}^{6} 4 n + 2$

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

$\sum_{n = 1}^{6} 4 n + 2 = 4 \sum_{n = 1}^{6} n + \sum_{n = 1}^{6} 2$

Evaluate $4 \sum_{n = 1}^{6} n$ .

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

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

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

$(4) (\frac{6 (6 + 1)}{2})$

Simplify.

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Simplify the expression.

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Add $6$ and $1$ .

$4 \frac{6 \cdot 7}{2}$

Multiply $6$ by $7$ .

$4 (\frac{42}{2})$

Cancel the common factor of $2$ .

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Factor $2$ out of $4$ .

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

Cancel the common factor.

$2 \cdot 2 \frac{42}{2}$

Rewrite the expression.

$2 \cdot 42$

Multiply $2$ by $42$ .

$84$

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

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The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Substitute the values into the formula.

$(2) (6)$

Multiply $2$ by $6$ .

$12$

Add the results of the summations.

$84 + 12$

Add $84$ and $12$ .

$96$