Calculus Examples

$\sum_{k = 3}^{9} - 9$

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

$\sum_{k = 3}^{9} - 9 = \sum_{k = 1}^{9} - 9 - \sum_{k = 1}^{2} - 9$

Evaluate $\sum_{k = 1}^{9} - 9$ .

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The formula for the summation of a constant is:

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

Substitute the values into the formula.

$(- 9) (9)$

Multiply $- 9$ by $9$ .

$- 81$

Evaluate $\sum_{k = 1}^{2} - 9$ .

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Expand the series for each value of $k$ .

$- 9 - 9$

Subtract $9$ from $- 9$ .

$- 18$

Replace the summations with the values found.

$- 81 + 18$

Add $- 81$ and $18$ .

$- 63$