Calculus Examples

$\sum_{k = 1}^{12} k^{2}$

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.

$\frac{12 (12 + 1) (2 \cdot 12 + 1)}{6}$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $12$ and $6$ .

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Step 3.1.1

Factor $6$ out of $12 (12 + 1) (2 \cdot 12 + 1)$ .

$\frac{6 (2 (12 + 1) (2 \cdot 12 + 1))}{6}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $6$ out of $6$ .

$\frac{6 (2 (12 + 1) (2 \cdot 12 + 1))}{6 (1)}$

Step 3.1.2.2

Cancel the common factor.

$\frac{6 (2 (12 + 1) (2 \cdot 12 + 1))}{6 \cdot 1}$

Step 3.1.2.3

Rewrite the expression.

$\frac{2 (12 + 1) (2 \cdot 12 + 1)}{1}$

Step 3.1.2.4

Divide $2 (12 + 1) (2 \cdot 12 + 1)$ by $1$ .

$2 (12 + 1) (2 \cdot 12 + 1)$

Step 3.2

Simplify the expression.

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Step 3.2.1

Add $12$ and $1$ .

$2 \cdot 13 (2 \cdot 12 + 1)$

Step 3.2.2

Multiply $2$ by $13$ .

$26 (2 \cdot 12 + 1)$

Step 3.2.3

Multiply $2$ by $12$ .

$26 (24 + 1)$

Step 3.2.4

Add $24$ and $1$ .

$26 \cdot 25$

Step 3.2.5

Multiply $26$ by $25$ .

$650$