Calculus Examples

$\sum_{i = 1}^{10} \frac{3}{8} n + \frac{3}{8}$

Step 1

Simplify the summation.

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Step 1.1

Combine $\frac{3}{8}$ and $n$ .

$\frac{3 n}{8} + \frac{3}{8}$

Step 1.2

Rewrite the summation.

$\sum_{i = 1}^{10} \frac{3 n}{8} + \frac{3}{8}$

Step 2

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

$\sum_{i = 1}^{10} \frac{3 n}{8} + \frac{3}{8} = \sum_{i = 1}^{10} \frac{3 n}{8} + \sum_{i = 1}^{10} \frac{3}{8}$

Step 3

Evaluate $\sum_{i = 1}^{10} \frac{3 n}{8}$ .

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Step 3.1

Factor $\frac{3}{8}$ out of the summation.

$(\frac{3}{8}) \sum_{k = 1}^{10} n$

Step 3.2

The formula for the summation of a constant is:

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

Step 3.3

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

$(\frac{3}{8}) ((n) (10))$

Step 3.4

Simplify.

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Step 3.4.1

Cancel the common factor of $2$ .

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Step 3.4.1.1

Factor $2$ out of $8$ .

$\frac{3}{2 (4)} ((n) (10))$

Step 3.4.1.2

Factor $2$ out of $(n) (10)$ .

$\frac{3}{2 (4)} (2 (n \cdot (5)))$

Step 3.4.1.3

Cancel the common factor.

$\frac{3}{2 \cdot 4} (2 (n \cdot (5)))$

Step 3.4.1.4

Rewrite the expression.

$\frac{3}{4} (n \cdot (5))$

Step 3.4.2

Combine $n$ and $\frac{3}{4}$ .

$\frac{n \cdot 3}{4} \cdot 5$

Step 3.4.3

Combine $\frac{n \cdot 3}{4}$ and $5$ .

$\frac{n \cdot 3 \cdot 5}{4}$

Step 3.4.4

Simplify the expression.

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Step 3.4.4.1

Multiply $5$ by $3$ .

$\frac{n \cdot 15}{4}$

Step 3.4.4.2

Move $15$ to the left of $n$ .

$\frac{15 n}{4}$

Step 4

Evaluate $\sum_{i = 1}^{10} \frac{3}{8}$ .

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Step 4.1

The formula for the summation of a constant is:

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

Step 4.2

Substitute the values into the formula.

$(\frac{3}{8}) (10)$

Step 4.3

Simplify.

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Factor $2$ out of $8$ .

$\frac{3}{2 (4)} \cdot 10$

Step 4.3.1.2

Factor $2$ out of $10$ .

$\frac{3}{2 \cdot 4} \cdot (2 \cdot 5)$

Step 4.3.1.3

Cancel the common factor.

$\frac{3}{2 \cdot 4} \cdot (2 \cdot 5)$

Step 4.3.1.4

Rewrite the expression.

$\frac{3}{4} \cdot 5$

Step 4.3.2

Combine $\frac{3}{4}$ and $5$ .

$\frac{3 \cdot 5}{4}$

Step 4.3.3

Multiply $3$ by $5$ .

$\frac{15}{4}$

Step 5

Add the results of the summations.

$\frac{15 n}{4} + \frac{15}{4}$