Calculus Examples

$\sum_{i = 1}^{12} i + 3$

Step 1

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

$\sum_{k = 1}^{12} i + 3 = \sum_{k = 1}^{12} i + \sum_{k = 1}^{12} 3$

Step 2

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

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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.

$\frac{12 (12 + 1)}{2}$

Step 2.3

Simplify.

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Step 2.3.1

Cancel the common factor of $12$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $12 (12 + 1)$ .

$\frac{2 (6 (12 + 1))}{2}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (6 (12 + 1))}{2 (1)}$

Step 2.3.1.2.2

Cancel the common factor.

$\frac{2 (6 (12 + 1))}{2 \cdot 1}$

Step 2.3.1.2.3

Rewrite the expression.

$\frac{6 (12 + 1)}{1}$

Step 2.3.1.2.4

Divide $6 (12 + 1)$ by $1$ .

$6 (12 + 1)$

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Add $12$ and $1$ .

$6 \cdot 13$

Step 2.3.2.2

Multiply $6$ by $13$ .

$78$

Step 3

Evaluate $\sum_{k = 1}^{12} 3$ .

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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.

$(3) (12)$

Step 3.3

Multiply $3$ by $12$ .

$36$

Step 4

Add the results of the summations.

$78 + 36$

Step 5

Add $78$ and $36$ .

$114$