Calculus Examples

$\sum_{k = 0}^{6} (\frac{3}{2}) \cdot k^{2}$

Step 1

Simplify the summation.

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Step 1.1

Combine $\frac{3}{2}$ and $k^{2}$ .

$\frac{3 k^{2}}{2}$

Step 1.2

Rewrite the summation.

$\sum_{k = 0}^{6} \frac{3 k^{2}}{2}$

Step 2

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

$\sum_{k = 0}^{6} \frac{3 k^{2}}{2} = \sum_{k = 1}^{6} \frac{3 k^{2}}{2} + \sum_{k = 0}^{0} \frac{3 k^{2}}{2}$

Step 3

Evaluate $\sum_{k = 1}^{6} \frac{3 k^{2}}{2}$ .

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Step 3.1

Factor $\frac{3}{2}$ out of the summation.

$(\frac{3}{2}) \sum_{k = 1}^{6} k^{2}$

Step 3.2

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 3.3

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

$(\frac{3}{2}) (\frac{6 (6 + 1) (2 \cdot 6 + 1)}{6})$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Add $6$ and $1$ .

$\frac{3}{2} \cdot \frac{6 \cdot 7 (2 \cdot 6 + 1)}{6}$

Step 3.4.1.2

Combine exponents.

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Step 3.4.1.2.1

Multiply $6$ by $7$ .

$\frac{3}{2} \cdot \frac{42 (2 \cdot 6 + 1)}{6}$

Step 3.4.1.2.2

Multiply $2$ by $6$ .

$\frac{3}{2} \cdot \frac{42 (12 + 1)}{6}$

Step 3.4.1.3

Add $12$ and $1$ .

$\frac{3}{2} \cdot \frac{42 \cdot 13}{6}$

Step 3.4.2

Reduce the expression by cancelling the common factors.

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Step 3.4.2.1

Multiply $42$ by $13$ .

$\frac{3}{2} \cdot \frac{546}{6}$

Step 3.4.2.2

Cancel the common factor of $3$ .

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Step 3.4.2.2.1

Factor $3$ out of $6$ .

$\frac{3}{2} \cdot \frac{546}{3 (2)}$

Step 3.4.2.2.2

Cancel the common factor.

$\frac{3}{2} \cdot \frac{546}{3 \cdot 2}$

Step 3.4.2.2.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{546}{2}$

Step 3.4.2.3

Cancel the common factor of $2$ .

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Step 3.4.2.3.1

Factor $2$ out of $546$ .

$\frac{1}{2} \cdot \frac{2 (273)}{2}$

Step 3.4.2.3.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot 273}{2}$

Step 3.4.2.3.3

Rewrite the expression.

$\frac{273}{2}$

Step 4

Evaluate $\sum_{k = 0}^{0} \frac{3 k^{2}}{2}$ .

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Step 4.1

Expand the series for each value of $k$ .

$\frac{3 \cdot 0^{2}}{2}$

Step 4.2

Simplify.

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Step 4.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{3 \cdot 0}{2}$

Step 4.2.2

Multiply $3$ by $0$ .

$\frac{0}{2}$

Step 4.2.3

Divide $0$ by $2$ .

$0$

Step 5

Replace the summations with the values found.

$\frac{273}{2} + 0$

Step 6

Add $\frac{273}{2}$ and $0$ .

$\frac{273}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{273}{2}$

Decimal Form:

$136.5$

Mixed Number Form:

$136 \frac{1}{2}$