Evaluate Using Summation Formulas sum from k=1 to 20 of 30-k^2

$\sum_{k = 1}^{20} 30 - k^{2}$

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

$\sum_{k = 1}^{20} 30 - k^{2} = \sum_{k = 1}^{20} 30 - 1 \sum_{k = 1}^{20} k^{2}$

Evaluate $\sum_{k = 1}^{20} 30$ .

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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 and make sure to multiply by the front term.

$(30) (20)$

Simplify.

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Remove parentheses.

$(30) (20)$

Multiply $30$ by $20$ .

$600$

Evaluate $1 \sum_{k = 1}^{20} k^{2}$ .

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The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

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

$- 1 \frac{20 (20 + 1) (2 \cdot 20 + 1)}{6}$

Simplify.

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Simplify the numerator.

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Add $20$ and $1$ .

$- 1 \frac{20 \cdot 21 (2 \cdot 20 + 1)}{6}$

Combine exponents.

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Multiply $20$ by $21$ .

$- 1 \frac{420 (2 \cdot 20 + 1)}{6}$

Multiply $2$ by $20$ .

$- 1 \frac{420 (40 + 1)}{6}$

Add $40$ and $1$ .

$- 1 \frac{420 \cdot 41}{6}$

Simplify the expression.

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Multiply $420$ by $41$ .

$- 1 (\frac{17220}{6})$

Divide $17220$ by $6$ .

$- 1 \cdot 2870$

Multiply $- 1$ by $2870$ .

$- 2870$

Add the results of the summations.

$600 - 2870$

Subtract $2870$ from $600$ .

$- 2270$

Evaluate Using Summation Formulas sum from k=1 to 20 of 30-k^2

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