Calculus Examples

$\sum_{n = 1}^{4} n^{2} + 2$

Step 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 2

Substitute the values into the formula.

$\frac{4 (4 + 1) (2 \cdot 4 + 1)}{6}$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $4$ and $6$ .

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Step 3.1.1

Factor $2$ out of $4 (4 + 1) (2 \cdot 4 + 1)$ .

$\frac{2 (2 (4 + 1) (2 \cdot 4 + 1))}{6}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $2$ out of $6$ .

$\frac{2 (2 (4 + 1) (2 \cdot 4 + 1))}{2 (3)}$

Step 3.1.2.2

Cancel the common factor.

$\frac{2 (2 (4 + 1) (2 \cdot 4 + 1))}{2 \cdot 3}$

Step 3.1.2.3

Rewrite the expression.

$\frac{2 (4 + 1) (2 \cdot 4 + 1)}{3}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Multiply $2$ by $4$ .

$\frac{2 (4 + 1) (8 + 1)}{3}$

Step 3.2.2

Add $4$ and $1$ .

$\frac{2 \cdot 5 (8 + 1)}{3}$

Step 3.2.3

Multiply $2$ by $5$ .

$\frac{10 (8 + 1)}{3}$

Step 3.2.4

Add $8$ and $1$ .

$\frac{10 \cdot 9}{3}$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $10$ by $9$ .

$\frac{90}{3}$

Step 3.3.2

Divide $90$ by $3$ .

$30$