Calculus Examples

$\sum_{i = 1}^{6} 6 i^{2}$

Step 1

Factor $6$ out of the summation.

$6 \sum_{k = 1}^{6} i^{2}$

Step 2

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

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Add $6$ and $1$ .

$6 \frac{6 \cdot 7 (2 \cdot 6 + 1)}{6}$

Step 4.1.2

Combine exponents.

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Step 4.1.2.1

Multiply $6$ by $7$ .

$6 \frac{42 (2 \cdot 6 + 1)}{6}$

Step 4.1.2.2

Multiply $2$ by $6$ .

$6 \frac{42 (12 + 1)}{6}$

Step 4.1.3

Add $12$ and $1$ .

$6 \frac{42 \cdot 13}{6}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Multiply $42$ by $13$ .

$6 (\frac{546}{6})$

Step 4.2.2

Cancel the common factor of $6$ .

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Step 4.2.2.1

Cancel the common factor.

$6 (\frac{546}{6})$

Step 4.2.2.2

Rewrite the expression.

$546$