Calculus Examples

$\sum_{n = 1}^{8} 1 + n^{2}$

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

$\sum_{n = 1}^{8} 1 + n^{2} = \sum_{n = 1}^{8} 1 + \sum_{n = 1}^{8} n^{2}$

Evaluate $\sum_{n = 1}^{8} 1$ .

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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.

$(1) (8)$

Multiply $8$ by $1$ .

$8$

Evaluate $\sum_{n = 1}^{8} n^{2}$ .

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The formula for the summation of a polynomial with degree $2$ is:

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

Substitute the values into the formula.

$\frac{8 (8 + 1) (2 \cdot 8 + 1)}{6}$

Simplify.

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Cancel the common factor of $8$ and $6$ .

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Factor $2$ out of $8 (8 + 1) (2 \cdot 8 + 1)$ .

$\frac{2 (4 (8 + 1) (2 \cdot 8 + 1))}{6}$

Cancel the common factors.

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Factor $2$ out of $6$ .

$\frac{2 (4 (8 + 1) (2 \cdot 8 + 1))}{2 (3)}$

Cancel the common factor.

$\frac{2 (4 (8 + 1) (2 \cdot 8 + 1))}{2 \cdot 3}$

Rewrite the expression.

$\frac{4 (8 + 1) (2 \cdot 8 + 1)}{3}$

Simplify the numerator.

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Multiply $2$ by $8$ .

$\frac{4 (8 + 1) (16 + 1)}{3}$

Add $8$ and $1$ .

$\frac{4 \cdot 9 (16 + 1)}{3}$

Multiply $4$ by $9$ .

$\frac{36 (16 + 1)}{3}$

Add $16$ and $1$ .

$\frac{36 \cdot 17}{3}$

Simplify the expression.

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Multiply $36$ by $17$ .

$\frac{612}{3}$

Divide $612$ by $3$ .

$204$

Add the results of the summations.

$8 + 204$

Add $8$ and $204$ .

$212$