Calculus Examples

$\sum_{k = 1}^{20} 5 k + 3$

Step 1

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

$\sum_{k = 1}^{20} 5 k + 3 = 5 \sum_{k = 1}^{20} k + \sum_{k = 1}^{20} 3$

Step 2

Evaluate $5 \sum_{k = 1}^{20} k$ .

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Step 2.1

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

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

Step 2.2

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

$(5) (\frac{20 (20 + 1)}{2})$

Step 2.3

Simplify.

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Step 2.3.1

Add $20$ and $1$ .

$5 \frac{20 \cdot 21}{2}$

Step 2.3.2

Multiply $20$ by $21$ .

$5 (\frac{420}{2})$

Step 2.3.3

Divide $420$ by $2$ .

$5 \cdot 210$

Step 2.3.4

Multiply $5$ by $210$ .

$1050$

Step 3

Evaluate $\sum_{k = 1}^{20} 3$ .

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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.

$(3) (20)$

Step 3.3

Multiply $3$ by $20$ .

$60$

Step 4

Add the results of the summations.

$1050 + 60$

Step 5

Add $1050$ and $60$ .

$1110$