Calculus Examples

$\sum_{n = 3}^{20} n (n + 1)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$n \cdot n + n \cdot 1$

Step 1.2

Multiply $n$ by $n$ .

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

Step 1.3

Multiply $n$ by $1$ .

$n^{2} + n$

Step 1.4

Rewrite the summation.

$\sum_{n = 3}^{20} n^{2} + n$

Step 2

Split the summation to make the starting value of $n$ equal to $1$ .

$\sum_{n = 3}^{20} n^{2} + n = \sum_{n = 1}^{20} n^{2} + n - \sum_{n = 1}^{2} n^{2} + n$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Substitute the values into the formula.

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

Step 3.2.3

Simplify.

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Step 3.2.3.1

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

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Step 3.2.3.1.1

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

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

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $2$ out of $6$ .

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

Step 3.2.3.1.2.2

Cancel the common factor.

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

Step 3.2.3.1.2.3

Rewrite the expression.

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

Step 3.2.3.2

Simplify the numerator.

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Step 3.2.3.2.1

Multiply $2$ by $20$ .

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

Step 3.2.3.2.2

Add $20$ and $1$ .

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

Step 3.2.3.2.3

Multiply $10$ by $21$ .

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

Step 3.2.3.2.4

Add $40$ and $1$ .

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

Step 3.2.3.3

Simplify the expression.

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Step 3.2.3.3.1

Multiply $210$ by $41$ .

$\frac{8610}{3}$

Step 3.2.3.3.2

Divide $8610$ by $3$ .

$2870$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Substitute the values into the formula.

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

Step 3.3.3

Simplify.

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Step 3.3.3.1

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

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Step 3.3.3.1.1

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

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

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $2$ out of $2$ .

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

Step 3.3.3.1.2.2

Cancel the common factor.

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

Step 3.3.3.1.2.3

Rewrite the expression.

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

Step 3.3.3.1.2.4

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

$10 (20 + 1)$

Step 3.3.3.2

Simplify the expression.

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Step 3.3.3.2.1

Add $20$ and $1$ .

$10 \cdot 21$

Step 3.3.3.2.2

Multiply $10$ by $21$ .

$210$

Step 3.4

Add the results of the summations.

$2870 + 210$

Step 3.5

Add $2870$ and $210$ .

$3080$

Step 4

Evaluate $\sum_{n = 1}^{2} n^{2} + n$ .

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Step 4.1

Expand the series for each value of $n$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Raise $1$ to the power of $2$ .

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

Step 4.2.2

Add $1$ and $1$ .

$2 + 2^{2} + 2$

Step 4.2.3

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

$2 + 4 + 2$

Step 4.2.4

Add $4$ and $2$ .

$2 + 6$

Step 4.2.5

Add $2$ and $6$ .

$8$

Step 5

Replace the summations with the values found.

$3080 - 8$

Step 6

Subtract $8$ from $3080$ .

$3072$