Calculus Examples

$\sum_{i = 1}^{16} 5 i - 4$

Step 1

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

$\sum_{k = 1}^{16} 5 i - 4 = 5 \sum_{k = 1}^{16} i + \sum_{k = 1}^{16} - 4$

Step 2

Evaluate $5 \sum_{k = 1}^{16} i$ .

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Step 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

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

$(5) (\frac{16 (16 + 1)}{2})$

Step 2.3

Simplify.

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Step 2.3.1

Add $16$ and $1$ .

$5 \frac{16 \cdot 17}{2}$

Step 2.3.2

Multiply $16$ by $17$ .

$5 (\frac{272}{2})$

Step 2.3.3

Divide $272$ by $2$ .

$5 \cdot 136$

Step 2.3.4

Multiply $5$ by $136$ .

$680$

Step 3

Evaluate $\sum_{k = 1}^{16} - 4$ .

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Step 3.1

The formula for the summation of a constant is:

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

Step 3.2

Substitute the values into the formula.

$(- 4) (16)$

Step 3.3

Multiply $- 4$ by $16$ .

$- 64$

Step 4

Add the results of the summations.

$680 - 64$

Step 5

Subtract $64$ from $680$ .

$616$