Calculus Examples

$\sum_{i = 1}^{9} 3 i^{2} + 8 i - 5$

Step 1

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

$\sum_{k = 1}^{9} 3 i^{2} + 8 i - 5 = 3 \sum_{k = 1}^{9} i^{2} + 8 \sum_{k = 1}^{9} i + \sum_{k = 1}^{9} - 5$

Step 2

Evaluate $3 \sum_{k = 1}^{9} i^{2}$ .

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Step 2.1

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 2.2

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

$(3) (\frac{9 (9 + 1) (2 \cdot 9 + 1)}{6})$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Add $9$ and $1$ .

$3 \frac{9 \cdot 10 (2 \cdot 9 + 1)}{6}$

Step 2.3.1.2

Combine exponents.

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Step 2.3.1.2.1

Multiply $9$ by $10$ .

$3 \frac{90 (2 \cdot 9 + 1)}{6}$

Step 2.3.1.2.2

Multiply $2$ by $9$ .

$3 \frac{90 (18 + 1)}{6}$

Step 2.3.1.3

Add $18$ and $1$ .

$3 \frac{90 \cdot 19}{6}$

Step 2.3.2

Reduce the expression by cancelling the common factors.

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Step 2.3.2.1

Multiply $90$ by $19$ .

$3 (\frac{1710}{6})$

Step 2.3.2.2

Cancel the common factor of $3$ .

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Step 2.3.2.2.1

Factor $3$ out of $6$ .

$3 \frac{1710}{3 (2)}$

Step 2.3.2.2.2

Cancel the common factor.

$3 \frac{1710}{3 \cdot 2}$

Step 2.3.2.2.3

Rewrite the expression.

$\frac{1710}{2}$

Step 2.3.2.3

Divide $1710$ by $2$ .

$855$

Step 3

Evaluate $8 \sum_{k = 1}^{9} i$ .

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Step 3.1

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

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

Step 3.2

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

$(8) (\frac{9 (9 + 1)}{2})$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the expression.

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Step 3.3.1.1

Add $9$ and $1$ .

$8 \frac{9 \cdot 10}{2}$

Step 3.3.1.2

Multiply $9$ by $10$ .

$8 (\frac{90}{2})$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Factor $2$ out of $8$ .

$2 (4) \frac{90}{2}$

Step 3.3.2.2

Cancel the common factor.

$2 \cdot 4 \frac{90}{2}$

Step 3.3.2.3

Rewrite the expression.

$4 \cdot 90$

Step 3.3.3

Multiply $4$ by $90$ .

$360$

Step 4

Evaluate $\sum_{k = 1}^{9} - 5$ .

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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.

$(- 5) (9)$

Step 4.3

Multiply $- 5$ by $9$ .

$- 45$

Step 5

Add the results of the summations.

$855 + 360 - 45$

Step 6

Simplify.

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Step 6.1

Add $855$ and $360$ .

$1215 - 45$

Step 6.2

Subtract $45$ from $1215$ .

$1170$