Calculus Examples

$\sum_{i = 7}^{16} 2 i - 12$

Step 1

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

$\sum_{k = 7}^{16} 2 i - 12 = \sum_{k = 1}^{16} 2 i - 12 - \sum_{k = 1}^{6} 2 i - 12$

Step 2

Evaluate $\sum_{k = 1}^{16} 2 i - 12$ .

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Step 2.1

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

$\sum_{k = 1}^{16} 2 i - 12 = 2 \sum_{k = 1}^{16} i + \sum_{k = 1}^{16} - 12$

Step 2.2

Evaluate $2 \sum_{k = 1}^{16} i$ .

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Step 2.2.1

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

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

Step 2.2.2

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

$(2) (\frac{16 (16 + 1)}{2})$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Simplify the expression.

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Step 2.2.3.1.1

Add $16$ and $1$ .

$2 \frac{16 \cdot 17}{2}$

Step 2.2.3.1.2

Multiply $16$ by $17$ .

$2 (\frac{272}{2})$

Step 2.2.3.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.1

Cancel the common factor.

$2 (\frac{272}{2})$

Step 2.2.3.2.2

Rewrite the expression.

$272$

Step 2.3

Evaluate $\sum_{k = 1}^{16} - 12$ .

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Step 2.3.1

The formula for the summation of a constant is:

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

Step 2.3.2

Substitute the values into the formula.

$(- 12) (16)$

Step 2.3.3

Multiply $- 12$ by $16$ .

$- 192$

Step 2.4

Add the results of the summations.

$272 - 192$

Step 2.5

Subtract $192$ from $272$ .

$80$

Step 3

Evaluate $\sum_{k = 1}^{6} 2 i - 12$ .

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Step 3.1

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

$\sum_{k = 1}^{6} 2 i - 12 = 2 \sum_{k = 1}^{6} i + \sum_{k = 1}^{6} - 12$

Step 3.2

Evaluate $2 \sum_{k = 1}^{6} i$ .

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Step 3.2.1

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

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

Step 3.2.2

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

$(2) (\frac{6 (6 + 1)}{2})$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Simplify the expression.

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Step 3.2.3.1.1

Add $6$ and $1$ .

$2 \frac{6 \cdot 7}{2}$

Step 3.2.3.1.2

Multiply $6$ by $7$ .

$2 (\frac{42}{2})$

Step 3.2.3.2

Cancel the common factor of $2$ .

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Step 3.2.3.2.1

Cancel the common factor.

$2 (\frac{42}{2})$

Step 3.2.3.2.2

Rewrite the expression.

$42$

Step 3.3

Evaluate $\sum_{k = 1}^{6} - 12$ .

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Step 3.3.1

The formula for the summation of a constant is:

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

Step 3.3.2

Substitute the values into the formula.

$(- 12) (6)$

Step 3.3.3

Multiply $- 12$ by $6$ .

$- 72$

Step 3.4

Add the results of the summations.

$42 - 72$

Step 3.5

Subtract $72$ from $42$ .

$- 30$

Step 4

Replace the summations with the values found.

$80 + 30$

Step 5

Add $80$ and $30$ .

$110$