Calculus Examples

$\sum_{j = 2}^{30} 6 j - \frac{4 j^{2}}{3}$

Step 1

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

$\sum_{j = 2}^{30} 6 j - \frac{4 j^{2}}{3} = \sum_{j = 1}^{30} 6 j - \frac{4 j^{2}}{3} - \sum_{j = 1}^{1} 6 j - \frac{4 j^{2}}{3}$

Step 2

Evaluate $\sum_{j = 1}^{30} 6 j - \frac{4 j^{2}}{3}$ .

Tap for more steps...

Step 2.1

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

$\sum_{j = 1}^{30} 6 j - \frac{4 j^{2}}{3} = 6 \sum_{j = 1}^{30} j + \sum_{j = 1}^{30} - \frac{4 j^{2}}{3}$

Step 2.2

Evaluate $6 \sum_{j = 1}^{30} j$ .

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

$(6) (\frac{30 (30 + 1)}{2})$

Step 2.2.3

Simplify.

Tap for more steps...

Step 2.2.3.1

Simplify the expression.

Tap for more steps...

Step 2.2.3.1.1

Add $30$ and $1$ .

$6 \frac{30 \cdot 31}{2}$

Step 2.2.3.1.2

Multiply $30$ by $31$ .

$6 (\frac{930}{2})$

Step 2.2.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.3.2.1

Factor $2$ out of $6$ .

$2 (3) \frac{930}{2}$

Step 2.2.3.2.2

Cancel the common factor.

$2 \cdot 3 \frac{930}{2}$

Step 2.2.3.2.3

Rewrite the expression.

$3 \cdot 930$

Step 2.2.3.3

Multiply $3$ by $930$ .

$2790$

Step 2.3

Evaluate $\sum_{j = 1}^{30} - \frac{4 j^{2}}{3}$ .

Tap for more steps...

Step 2.3.1

Factor $- \frac{4}{3}$ out of the summation.

$(- \frac{4}{3}) \sum_{j = 1}^{30} j^{2}$

Step 2.3.2

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 2.3.3

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

$(- \frac{4}{3}) (\frac{30 (30 + 1) (2 \cdot 30 + 1)}{6})$

Step 2.3.4

Simplify.

Tap for more steps...

Step 2.3.4.1

Simplify the numerator.

Tap for more steps...

Step 2.3.4.1.1

Add $30$ and $1$ .

$- \frac{4}{3} \cdot \frac{30 \cdot 31 (2 \cdot 30 + 1)}{6}$

Step 2.3.4.1.2

Combine exponents.

Tap for more steps...

Step 2.3.4.1.2.1

Multiply $30$ by $31$ .

$- \frac{4}{3} \cdot \frac{930 (2 \cdot 30 + 1)}{6}$

Step 2.3.4.1.2.2

Multiply $2$ by $30$ .

$- \frac{4}{3} \cdot \frac{930 (60 + 1)}{6}$

Step 2.3.4.1.3

Add $60$ and $1$ .

$- \frac{4}{3} \cdot \frac{930 \cdot 61}{6}$

Step 2.3.4.2

Simplify terms.

Tap for more steps...

Step 2.3.4.2.1

Multiply $930$ by $61$ .

$- \frac{4}{3} \cdot \frac{56730}{6}$

Step 2.3.4.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.4.2.2.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{- 4}{3} \cdot \frac{56730}{6}$

Step 2.3.4.2.2.2

Factor $2$ out of $- 4$ .

$\frac{2 (- 2)}{3} \cdot \frac{56730}{6}$

Step 2.3.4.2.2.3

Factor $2$ out of $6$ .

$\frac{2 \cdot - 2}{3} \cdot \frac{56730}{2 \cdot 3}$

Step 2.3.4.2.2.4

Cancel the common factor.

$\frac{2 \cdot - 2}{3} \cdot \frac{56730}{2 \cdot 3}$

Step 2.3.4.2.2.5

Rewrite the expression.

$\frac{- 2}{3} \cdot \frac{56730}{3}$

Step 2.3.4.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.4.2.3.1

Factor $3$ out of $56730$ .

$\frac{- 2}{3} \cdot \frac{3 (18910)}{3}$

Step 2.3.4.2.3.2

Cancel the common factor.

$\frac{- 2}{3} \cdot \frac{3 \cdot 18910}{3}$

Step 2.3.4.2.3.3

Rewrite the expression.

$- 2 (\frac{18910}{3})$

Step 2.3.4.2.4

Combine $- 2$ and $\frac{18910}{3}$ .

$\frac{- 2 \cdot 18910}{3}$

Step 2.3.4.2.5

Simplify the expression.

Tap for more steps...

Step 2.3.4.2.5.1

Multiply $- 2$ by $18910$ .

$\frac{- 37820}{3}$

Step 2.3.4.2.5.2

Move the negative in front of the fraction.

$- \frac{37820}{3}$

Step 2.4

Add the results of the summations.

$2790 - \frac{37820}{3}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

To write $2790$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2790 \cdot \frac{3}{3} - \frac{37820}{3}$

Step 2.5.2

Combine $2790$ and $\frac{3}{3}$ .

$\frac{2790 \cdot 3}{3} - \frac{37820}{3}$

Step 2.5.3

Combine the numerators over the common denominator.

$\frac{2790 \cdot 3 - 37820}{3}$

Step 2.5.4

Simplify the numerator.

Tap for more steps...

Step 2.5.4.1

Multiply $2790$ by $3$ .

$\frac{8370 - 37820}{3}$

Step 2.5.4.2

Subtract $37820$ from $8370$ .

$\frac{- 29450}{3}$

Step 2.5.5

Move the negative in front of the fraction.

$- \frac{29450}{3}$

Step 3

Evaluate $\sum_{j = 1}^{1} 6 j - \frac{4 j^{2}}{3}$ .

Tap for more steps...

Step 3.1

Expand the series for each value of $j$ .

$6 \cdot 1 - \frac{4 \cdot 1^{2}}{3}$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $6$ by $1$ .

$6 - \frac{4 \cdot 1^{2}}{3}$

Step 3.2.1.2

One to any power is one.

$6 - \frac{4 \cdot 1}{3}$

Step 3.2.1.3

Multiply $4$ by $1$ .

$6 - \frac{4}{3}$

Step 3.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 \cdot \frac{3}{3} - \frac{4}{3}$

Step 3.2.3

Combine $6$ and $\frac{3}{3}$ .

$\frac{6 \cdot 3}{3} - \frac{4}{3}$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{6 \cdot 3 - 4}{3}$

Step 3.2.5

Simplify the numerator.

Tap for more steps...

Step 3.2.5.1

Multiply $6$ by $3$ .

$\frac{18 - 4}{3}$

Step 3.2.5.2

Subtract $4$ from $18$ .

$\frac{14}{3}$

Step 4

Replace the summations with the values found.

$- \frac{29450}{3} - \frac{14}{3}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine the numerators over the common denominator.

$\frac{- 29450 - 14}{3}$

Step 5.2

Simplify the expression.

Tap for more steps...

Step 5.2.1

Subtract $14$ from $- 29450$ .

$\frac{- 29464}{3}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{29464}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{29464}{3}$

Decimal Form:

$- 9821.\overline{3}$

Mixed Number Form:

$- 9821 \frac{1}{3}$