Calculus Examples

$5 \sum_{i = 1}^{14} i^{2} + 14 \cdot 13$

Step 1

Simplify the summation.

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Step 1.1

Multiply $14$ by $13$ .

$i^{2} + 182$

Step 1.2

Rewrite the summation.

$5 \sum_{i = 1}^{14} i^{2} + 182$

Step 2

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

$5 \sum_{i = 1}^{14} i^{2} + 182 = 5 \sum_{i = 1}^{14} i^{2} + 5 \sum_{i = 1}^{14} 182$

Step 3

Evaluate $5 \sum_{i = 1}^{14} i^{2}$ .

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Step 3.1

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

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

Step 3.2

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

$(5) (\frac{14 (14 + 1) (2 \cdot 14 + 1)}{6})$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Add $14$ and $1$ .

$5 \frac{14 \cdot 15 (2 \cdot 14 + 1)}{6}$

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Multiply $14$ by $15$ .

$5 \frac{210 (2 \cdot 14 + 1)}{6}$

Step 3.3.1.2.2

Multiply $2$ by $14$ .

$5 \frac{210 (28 + 1)}{6}$

Step 3.3.1.3

Add $28$ and $1$ .

$5 \frac{210 \cdot 29}{6}$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $210$ by $29$ .

$5 (\frac{6090}{6})$

Step 3.3.2.2

Divide $6090$ by $6$ .

$5 \cdot 1015$

Step 3.3.2.3

Multiply $5$ by $1015$ .

$5075$

Step 4

Evaluate $5 \sum_{i = 1}^{14} 182$ .

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Step 4.1

The formula for the summation of a constant is:

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

Step 4.2

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

$(5) ((182) (14))$

Step 4.3

Multiply $(5) ((182) (14))$ .

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Step 4.3.1

Multiply $182$ by $14$ .

$5 \cdot 2548$

Step 4.3.2

Multiply $5$ by $2548$ .

$12740$

Step 5

Add the results of the summations.

$5075 + 12740$

Step 6

Add $5075$ and $12740$ .

$17815$