Calculus Examples

$\sum_{i = 1}^{20} i - 1^{2}$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

One to any power is one.

$i - 1 \cdot 1$

Step 1.1.2

Multiply $- 1$ by $1$ .

$i - 1$

Step 1.2

Rewrite the summation.

$\sum_{i = 1}^{20} i - 1$

Step 2

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

$\sum_{i = 1}^{20} i - 1 = \sum_{i = 1}^{20} i + \sum_{i = 1}^{20} - 1$

Step 3

Evaluate $\sum_{i = 1}^{20} i$ .

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Step 3.1

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

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

Step 3.2

Substitute the values into the formula.

$\frac{20 (20 + 1)}{2}$

Step 3.3

Simplify.

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Step 3.3.1

Cancel the common factor of $20$ and $2$ .

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Step 3.3.1.1

Factor $2$ out of $20 (20 + 1)$ .

$\frac{2 (10 (20 + 1))}{2}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (10 (20 + 1))}{2 (1)}$

Step 3.3.1.2.2

Cancel the common factor.

$\frac{2 (10 (20 + 1))}{2 \cdot 1}$

Step 3.3.1.2.3

Rewrite the expression.

$\frac{10 (20 + 1)}{1}$

Step 3.3.1.2.4

Divide $10 (20 + 1)$ by $1$ .

$10 (20 + 1)$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Add $20$ and $1$ .

$10 \cdot 21$

Step 3.3.2.2

Multiply $10$ by $21$ .

$210$

Step 4

Evaluate $\sum_{i = 1}^{20} - 1$ .

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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.

$(- 1) (20)$

Step 4.3

Multiply $- 1$ by $20$ .

$- 20$

Step 5

Add the results of the summations.

$210 - 20$

Step 6

Subtract $20$ from $210$ .

$190$