Calculus Examples

$\sum_{i = 1}^{100} \frac{i + 1}{10000}$

Step 1

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

$\sum_{k = 1}^{100} \frac{i}{10000} + \frac{1}{10000} = \sum_{k = 1}^{100} \frac{i}{10000} + \sum_{k = 1}^{100} \frac{1}{10000}$

Step 2

Evaluate $\sum_{k = 1}^{100} \frac{i}{10000}$ .

Tap for more steps...

Step 2.1

Factor $\frac{1}{10000}$ out of the summation.

$\frac{1}{10000} \sum_{k = 1}^{100} i$

Step 2.2

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

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

Step 2.3

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

$(\frac{1}{10000}) (\frac{100 (100 + 1)}{2})$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Simplify the expression.

Tap for more steps...

Step 2.4.1.1

Add $100$ and $1$ .

$\frac{1}{10000} \cdot \frac{100 \cdot 101}{2}$

Step 2.4.1.2

Multiply $100$ by $101$ .

$\frac{1}{10000} \cdot \frac{10100}{2}$

Step 2.4.2

Cancel the common factor of $100$ .

Tap for more steps...

Step 2.4.2.1

Factor $100$ out of $10000$ .

$\frac{1}{100 (100)} \cdot \frac{10100}{2}$

Step 2.4.2.2

Factor $100$ out of $10100$ .

$\frac{1}{100 \cdot 100} \cdot \frac{100 \cdot 101}{2}$

Step 2.4.2.3

Cancel the common factor.

$\frac{1}{100 \cdot 100} \cdot \frac{100 \cdot 101}{2}$

Step 2.4.2.4

Rewrite the expression.

$\frac{1}{100} \cdot \frac{101}{2}$

Step 2.4.3

Multiply $\frac{1}{100}$ by $\frac{101}{2}$ .

$\frac{101}{100 \cdot 2}$

Step 2.4.4

Multiply $100$ by $2$ .

$\frac{101}{200}$

Step 3

Evaluate $\sum_{k = 1}^{100} \frac{1}{10000}$ .

Tap for more steps...

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.

$(\frac{1}{10000}) (100)$

Step 3.3

Cancel the common factor of $100$ .

Tap for more steps...

Step 3.3.1

Factor $100$ out of $10000$ .

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

Step 3.3.2

Cancel the common factor.

$\frac{1}{100 \cdot 100} \cdot 100$

Step 3.3.3

Rewrite the expression.

$\frac{1}{100}$

Step 4

Add the results of the summations.

$\frac{101}{200} + \frac{1}{100}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

To write $\frac{1}{100}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{101}{200} + \frac{1}{100} \cdot \frac{2}{2}$

Step 5.2

Write each expression with a common denominator of $200$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.2.1

Multiply $\frac{1}{100}$ by $\frac{2}{2}$ .

$\frac{101}{200} + \frac{2}{100 \cdot 2}$

Step 5.2.2

Multiply $100$ by $2$ .

$\frac{101}{200} + \frac{2}{200}$

Step 5.3

Combine the numerators over the common denominator.

$\frac{101 + 2}{200}$

Step 5.4

Add $101$ and $2$ .

$\frac{103}{200}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{103}{200}$

Decimal Form:

$0.515$