Calculus Examples

$\sum_{i = 1}^{10} \frac{2 i + 1}{10^{2}}$

Step 1

Simplify the summation.

Tap for more steps...

Step 1.1

Raise $10$ to the power of $2$ .

$\frac{2 i + 1}{100}$

Step 1.2

Rewrite the summation.

$\sum_{i = 1}^{10} \frac{2 i + 1}{100}$

Step 2

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

$\sum_{i = 1}^{10} \frac{i}{50} + \frac{1}{100} = \sum_{i = 1}^{10} \frac{i}{50} + \sum_{i = 1}^{10} \frac{1}{100}$

Step 3

Evaluate $\sum_{i = 1}^{10} \frac{i}{50}$ .

Tap for more steps...

Step 3.1

Factor $\frac{1}{50}$ out of the summation.

$\frac{1}{50} \sum_{k = 1}^{10} i$

Step 3.2

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

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

Step 3.3

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

$(\frac{1}{50}) (\frac{10 (10 + 1)}{2})$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Simplify the expression.

Tap for more steps...

Step 3.4.1.1

Add $10$ and $1$ .

$\frac{1}{50} \cdot \frac{10 \cdot 11}{2}$

Step 3.4.1.2

Multiply $10$ by $11$ .

$\frac{1}{50} \cdot \frac{110}{2}$

Step 3.4.2

Cancel the common factor of $10$ .

Tap for more steps...

Step 3.4.2.1

Factor $10$ out of $50$ .

$\frac{1}{10 (5)} \cdot \frac{110}{2}$

Step 3.4.2.2

Factor $10$ out of $110$ .

$\frac{1}{10 \cdot 5} \cdot \frac{10 \cdot 11}{2}$

Step 3.4.2.3

Cancel the common factor.

$\frac{1}{10 \cdot 5} \cdot \frac{10 \cdot 11}{2}$

Step 3.4.2.4

Rewrite the expression.

$\frac{1}{5} \cdot \frac{11}{2}$

Step 3.4.3

Multiply $\frac{1}{5}$ by $\frac{11}{2}$ .

$\frac{11}{5 \cdot 2}$

Step 3.4.4

Multiply $5$ by $2$ .

$\frac{11}{10}$

Step 4

Evaluate $\sum_{i = 1}^{10} \frac{1}{100}$ .

Tap for more steps...

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.

$(\frac{1}{100}) (10)$

Step 4.3

Cancel the common factor of $10$ .

Tap for more steps...

Step 4.3.1

Factor $10$ out of $100$ .

$\frac{1}{10 (10)} \cdot 10$

Step 4.3.2

Cancel the common factor.

$\frac{1}{10 \cdot 10} \cdot 10$

Step 4.3.3

Rewrite the expression.

$\frac{1}{10}$

Step 5

Add the results of the summations.

$\frac{11}{10} + \frac{1}{10}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Combine the numerators over the common denominator.

$\frac{11 + 1}{10}$

Step 6.2

Add $11$ and $1$ .

$\frac{12}{10}$

Step 6.3

Cancel the common factor of $12$ and $10$ .

Tap for more steps...

Step 6.3.1

Factor $2$ out of $12$ .

$\frac{2 (6)}{10}$

Step 6.3.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 6}{2 \cdot 5}$

Step 6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 6}{2 \cdot 5}$

Step 6.3.2.3

Rewrite the expression.

$\frac{6}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{6}{5}$

Decimal Form:

$1.2$

Mixed Number Form:

$1 \frac{1}{5}$