Calculus Examples

$\sum_{i = 1}^{20} {(i + 1)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(i + 1)}^{2}$ as $(i + 1) (i + 1)$ .

$(i + 1) (i + 1)$

Step 1.2

Expand $(i + 1) (i + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$i (i + 1) + 1 (i + 1)$

Step 1.2.2

Apply the distributive property.

$i \cdot i + i \cdot 1 + 1 (i + 1)$

Step 1.2.3

Apply the distributive property.

$i \cdot i + i \cdot 1 + 1 i + 1 \cdot 1$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $i$ by $i$ .

$i^{2} + i \cdot 1 + 1 i + 1 \cdot 1$

Step 1.3.1.2

Multiply $i$ by $1$ .

$i^{2} + i + 1 i + 1 \cdot 1$

Step 1.3.1.3

Multiply $i$ by $1$ .

$i^{2} + i + i + 1 \cdot 1$

Step 1.3.1.4

Multiply $1$ by $1$ .

$i^{2} + i + i + 1$

Step 1.3.2

Add $i$ and $i$ .

$i^{2} + 2 i + 1$

Step 1.4

Rewrite the summation.

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

Step 2

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

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

Step 3

Evaluate $\sum_{i = 1}^{20} i^{2}$ .

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Step 3.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} i^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 3.2

Substitute the values into the formula.

$\frac{20 (20 + 1) (2 \cdot 20 + 1)}{6}$

Step 3.3

Simplify.

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Step 3.3.1

Cancel the common factor of $20$ and $6$ .

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Step 3.3.1.1

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

$\frac{2 (10 (20 + 1) (2 \cdot 20 + 1))}{6}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $6$ .

$\frac{2 (10 (20 + 1) (2 \cdot 20 + 1))}{2 (3)}$

Step 3.3.1.2.2

Cancel the common factor.

$\frac{2 (10 (20 + 1) (2 \cdot 20 + 1))}{2 \cdot 3}$

Step 3.3.1.2.3

Rewrite the expression.

$\frac{10 (20 + 1) (2 \cdot 20 + 1)}{3}$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Multiply $2$ by $20$ .

$\frac{10 (20 + 1) (40 + 1)}{3}$

Step 3.3.2.2

Add $20$ and $1$ .

$\frac{10 \cdot 21 (40 + 1)}{3}$

Step 3.3.2.3

Multiply $10$ by $21$ .

$\frac{210 (40 + 1)}{3}$

Step 3.3.2.4

Add $40$ and $1$ .

$\frac{210 \cdot 41}{3}$

Step 3.3.3

Simplify the expression.

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Step 3.3.3.1

Multiply $210$ by $41$ .

$\frac{8610}{3}$

Step 3.3.3.2

Divide $8610$ by $3$ .

$2870$

Step 4

Evaluate $2 \sum_{i = 1}^{20} i$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify.

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Step 4.3.1

Simplify the expression.

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Step 4.3.1.1

Add $20$ and $1$ .

$2 \frac{20 \cdot 21}{2}$

Step 4.3.1.2

Multiply $20$ by $21$ .

$2 (\frac{420}{2})$

Step 4.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.1

Cancel the common factor.

$2 (\frac{420}{2})$

Step 4.3.2.2

Rewrite the expression.

$420$

Step 5

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

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Step 5.1

The formula for the summation of a constant is:

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

Step 5.2

Substitute the values into the formula.

$(1) (20)$

Step 5.3

Multiply $20$ by $1$ .

$20$

Step 6

Add the results of the summations.

$2870 + 420 + 20$

Step 7

Simplify.

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Step 7.1

Add $2870$ and $420$ .

$3290 + 20$

Step 7.2

Add $3290$ and $20$ .

$3310$