Calculus Examples

$\sum_{n = 3}^{8} 5 n$

Step 1

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

$\sum_{n = 3}^{8} 5 n = \sum_{n = 1}^{8} 5 n - \sum_{n = 1}^{2} 5 n$

Step 2

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

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Step 2.1

Factor $5$ out of the summation.

$(5) \sum_{n = 1}^{8} n$

Step 2.2

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

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

$(5) (\frac{8 (8 + 1)}{2})$

Step 2.4

Simplify.

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Step 2.4.1

Add $8$ and $1$ .

$5 \frac{8 \cdot 9}{2}$

Step 2.4.2

Multiply $8$ by $9$ .

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

Step 2.4.3

Divide $72$ by $2$ .

$5 \cdot 36$

Step 2.4.4

Multiply $5$ by $36$ .

$180$

Step 3

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

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Step 3.1

Expand the series for each value of $n$ .

$5 \cdot 1 + 5 \cdot 2$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $5$ by $1$ .

$5 + 5 \cdot 2$

Step 3.2.2

Multiply $5$ by $2$ .

$5 + 10$

Step 3.2.3

Add $5$ and $10$ .

$15$

Step 4

Replace the summations with the values found.

$180 - 15$

Step 5

Subtract $15$ from $180$ .

$165$