Evaluate sum from n=3 to 8 of 5n

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

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

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

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

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Factor $5$ out of the summation.

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

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

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

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

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

Simplify.

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Add $8$ and $1$ .

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

Multiply $8$ by $9$ .

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

Divide $72$ by $2$ .

$5 \cdot 36$

Multiply $5$ by $36$ .

$180$

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

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Expand the series for each value of $n$ .

$5 (1) + 5 (2)$

Simplify.

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Simplify each term.

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Multiply $5$ by $1$ .

$5 + 5 (2)$

Multiply $5$ by $2$ .

$5 + 10$

Add $5$ and $10$ .

$15$

Replace the summations with the values found.

$180 - 15$

Subtract $15$ from $180$ .

$165$

Evaluate sum from n=3 to 8 of 5n

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