Calculus Examples

$3 \sum_{p = 3}^{8} 5 - p$

Step 1

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

$3 \sum_{p = 3}^{8} 5 - p = 3 (\sum_{p = 1}^{8} 5 - p - \sum_{p = 1}^{2} 5 - p)$

Step 2

Evaluate $\sum_{p = 1}^{8} 5 - p$ .

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Step 2.1

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

$\sum_{p = 1}^{8} 5 - p = \sum_{p = 1}^{8} 5 - 1 \sum_{p = 1}^{8} p$

Step 2.2

Evaluate $\sum_{p = 1}^{8} 5$ .

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Step 2.2.1

The formula for the summation of a constant is:

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

Step 2.2.2

Substitute the values into the formula.

$(5) (8)$

Step 2.2.3

Multiply $5$ by $8$ .

$40$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Add $8$ and $1$ .

$- 1 \frac{8 \cdot 9}{2}$

Step 2.3.3.2

Multiply $8$ by $9$ .

$- 1 (\frac{72}{2})$

Step 2.3.3.3

Divide $72$ by $2$ .

$- 1 \cdot 36$

Step 2.3.3.4

Multiply $- 1$ by $36$ .

$- 36$

Step 2.4

Add the results of the summations.

$40 - 36$

Step 2.5

Subtract $36$ from $40$ .

$4$

Step 3

Evaluate $\sum_{p = 1}^{2} 5 - p$ .

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Step 3.1

Expand the series for each value of $p$ .

$5 - 1 + 5 - 2$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $5$ .

$4 + 5 - 2$

Step 3.2.2

Subtract $2$ from $5$ .

$4 + 3$

Step 3.2.3

Add $4$ and $3$ .

$7$

Step 4

Replace the summations with the values found.

$3 (4 - 7)$

Step 5

Simplify.

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Step 5.1

Subtract $7$ from $4$ .

$3 \cdot - 3$

Step 5.2

Multiply $3$ by $- 3$ .

$- 9$