Calculus Examples

$\sum_{n = 1}^{100} n$

Step 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

Substitute the values into the formula.

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $100$ and $2$ .

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Step 3.1.1

Factor $2$ out of $100 (100 + 1)$ .

$\frac{2 (50 (100 + 1))}{2}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (50 (100 + 1))}{2 (1)}$

Step 3.1.2.2

Cancel the common factor.

$\frac{2 (50 (100 + 1))}{2 \cdot 1}$

Step 3.1.2.3

Rewrite the expression.

$\frac{50 (100 + 1)}{1}$

Step 3.1.2.4

Divide $50 (100 + 1)$ by $1$ .

$50 (100 + 1)$

$50 (100 + 1)$

$50 (100 + 1)$

Step 3.2

Simplify the expression.

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Step 3.2.1

Add $100$ and $1$ .

$50 \cdot 101$

Step 3.2.2

Multiply $50$ by $101$ .

$5050$

$5050$

$5050$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$