Evaluate sum from i=2 to 7 of 3i-15

$\sum_{i = 2}^{7} 3 i - 15$

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

$\sum_{i = 2}^{7} 3 i - 15 = \sum_{i = 1}^{7} 3 i - 15 - \sum_{i = 1}^{1} 3 i - 15$

Evaluate $\sum_{i = 1}^{7} 3 i - 15$ .

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

$\sum_{i = 1}^{7} 3 i - 15 = 3 \sum_{i = 1}^{7} i + \sum_{i = 1}^{7} - 15$

Evaluate $3 \sum_{i = 1}^{7} i$ .

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

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

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

$(3) (\frac{7 (7 + 1)}{2})$

Simplify.

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Add $7$ and $1$ .

$3 \frac{7 \cdot 8}{2}$

Multiply $7$ by $8$ .

$3 (\frac{56}{2})$

Divide $56$ by $2$ .

$3 \cdot 28$

Multiply $3$ by $28$ .

$84$

Evaluate $\sum_{i = 1}^{7} - 15$ .

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The formula for the summation of a constant is:

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

Substitute the values into the formula.

$(- 15) (7)$

Simplify.

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Remove parentheses.

$(- 15) (7)$

Multiply $- 15$ by $7$ .

$- 105$

Add the results of the summations.

$84 - 105$

Subtract $105$ from $84$ .

$- 21$

Evaluate $\sum_{i = 1}^{1} 3 i - 15$ .

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Expand the series for each value of $i$ .

$3 (1) - 15$

Simplify.

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Multiply $3$ by $1$ .

$3 - 15$

Subtract $15$ from $3$ .

$- 12$

Replace the summations with the values found.

$- 21 + 12$

Add $- 21$ and $12$ .

$- 9$

Evaluate sum from i=2 to 7 of 3i-15

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