Calculus Examples

$\sum_{i = 0}^{8} 10 i$

Step 1

Split the summation to make the starting value of $i$ equal to $1$ .

$\sum_{k = 0}^{8} 10 i = \sum_{k = 1}^{8} 10 i + \sum_{k = 0}^{0} 10 i$

Step 2

Evaluate $\sum_{k = 1}^{8} 10 i$ .

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Step 2.1

Factor $10$ out of the summation.

$10 \sum_{k = 1}^{8} i$

Step 2.2

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

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

Step 2.3

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

$(10) (\frac{8 (8 + 1)}{2})$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the expression.

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Step 2.4.1.1

Add $8$ and $1$ .

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

Step 2.4.1.2

Multiply $8$ by $9$ .

$10 (\frac{72}{2})$

Step 2.4.2

Cancel the common factor of $2$ .

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Step 2.4.2.1

Factor $2$ out of $10$ .

$2 (5) \frac{72}{2}$

Step 2.4.2.2

Cancel the common factor.

$2 \cdot 5 \frac{72}{2}$

Step 2.4.2.3

Rewrite the expression.

$5 \cdot 72$

Step 2.4.3

Multiply $5$ by $72$ .

$360$

Step 3

Evaluate $\sum_{k = 0}^{0} 10 i$ .

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Step 3.1

Expand the series for each value of $i$ .

$10 \cdot 0$

Step 3.2

Multiply $10$ by $0$ .

$0$

Step 4

Replace the summations with the values found.

$360 + 0$

Step 5

Add $360$ and $0$ .

$360$