Calculus Examples

$\sum_{n = 1}^{4} \frac{5 n}{n + 3}$

Step 1

Expand the series for each value of $n$ .

$\frac{5 \cdot 1}{1 + 3} + \frac{5 \cdot 2}{2 + 3} + \frac{5 \cdot 3}{3 + 3} + \frac{5 \cdot 4}{4 + 3}$

Step 2

Simplify.

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Step 2.1

Multiply $5$ by $1$ .

$\frac{5}{1 + 3} + \frac{5 \cdot 2}{2 + 3} + \frac{5 \cdot 3}{3 + 3} + \frac{5 \cdot 4}{4 + 3}$

Step 2.2

Multiply $5$ by $2$ .

$\frac{5}{1 + 3} + \frac{10}{2 + 3} + \frac{5 \cdot 3}{3 + 3} + \frac{5 \cdot 4}{4 + 3}$

Step 2.3

Multiply $5$ by $3$ .

$\frac{5}{1 + 3} + \frac{10}{2 + 3} + \frac{15}{3 + 3} + \frac{5 \cdot 4}{4 + 3}$

Step 2.4

Multiply $5$ by $4$ .

$\frac{5}{1 + 3} + \frac{10}{2 + 3} + \frac{15}{3 + 3} + \frac{20}{4 + 3}$

Step 2.5

Simplify each term.

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Step 2.5.1

Add $1$ and $3$ .

$\frac{5}{4} + \frac{10}{2 + 3} + \frac{15}{3 + 3} + \frac{20}{4 + 3}$

Step 2.5.2

Add $2$ and $3$ .

$\frac{5}{4} + \frac{10}{5} + \frac{15}{3 + 3} + \frac{20}{4 + 3}$

Step 2.5.3

Divide $10$ by $5$ .

$\frac{5}{4} + 2 + \frac{15}{3 + 3} + \frac{20}{4 + 3}$

Step 2.5.4

Cancel the common factor of $15$ and $3 + 3$ .

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Step 2.5.4.1

Factor $3$ out of $15$ .

$\frac{5}{4} + 2 + \frac{3 \cdot 5}{3 + 3} + \frac{20}{4 + 3}$

Step 2.5.4.2

Cancel the common factors.

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Step 2.5.4.2.1

Factor $3$ out of $3$ .

$\frac{5}{4} + 2 + \frac{3 \cdot 5}{3 (1) + 3} + \frac{20}{4 + 3}$

Step 2.5.4.2.2

Factor $3$ out of $3$ .

$\frac{5}{4} + 2 + \frac{3 \cdot 5}{3 \cdot 1 + 3 \cdot 1} + \frac{20}{4 + 3}$

Step 2.5.4.2.3

Factor $3$ out of $3 \cdot 1 + 3 \cdot 1$ .

$\frac{5}{4} + 2 + \frac{3 \cdot 5}{3 \cdot (1 + 1)} + \frac{20}{4 + 3}$

Step 2.5.4.2.4

Cancel the common factor.

$\frac{5}{4} + 2 + \frac{3 \cdot 5}{3 \cdot (1 + 1)} + \frac{20}{4 + 3}$

Step 2.5.4.2.5

Rewrite the expression.

$\frac{5}{4} + 2 + \frac{5}{1 + 1} + \frac{20}{4 + 3}$

Step 2.5.5

Add $1$ and $1$ .

$\frac{5}{4} + 2 + \frac{5}{2} + \frac{20}{4 + 3}$

Step 2.5.6

Add $4$ and $3$ .

$\frac{5}{4} + 2 + \frac{5}{2} + \frac{20}{7}$

Step 2.6

Find the common denominator.

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Step 2.6.1

Multiply $\frac{5}{4}$ by $\frac{7}{7}$ .

$\frac{5}{4} \cdot \frac{7}{7} + 2 + \frac{5}{2} + \frac{20}{7}$

Step 2.6.2

Multiply $\frac{5}{4}$ by $\frac{7}{7}$ .

$\frac{5 \cdot 7}{4 \cdot 7} + 2 + \frac{5}{2} + \frac{20}{7}$

Step 2.6.3

Write $2$ as a fraction with denominator $1$ .

$\frac{5 \cdot 7}{4 \cdot 7} + \frac{2}{1} + \frac{5}{2} + \frac{20}{7}$

Step 2.6.4

Multiply $\frac{2}{1}$ by $\frac{28}{28}$ .

$\frac{5 \cdot 7}{4 \cdot 7} + \frac{2}{1} \cdot \frac{28}{28} + \frac{5}{2} + \frac{20}{7}$

Step 2.6.5

Multiply $\frac{2}{1}$ by $\frac{28}{28}$ .

$\frac{5 \cdot 7}{4 \cdot 7} + \frac{2 \cdot 28}{28} + \frac{5}{2} + \frac{20}{7}$

Step 2.6.6

Multiply $\frac{5}{2}$ by $\frac{14}{14}$ .

$\frac{5 \cdot 7}{4 \cdot 7} + \frac{2 \cdot 28}{28} + \frac{5}{2} \cdot \frac{14}{14} + \frac{20}{7}$

Step 2.6.7

Multiply $\frac{5}{2}$ by $\frac{14}{14}$ .

$\frac{5 \cdot 7}{4 \cdot 7} + \frac{2 \cdot 28}{28} + \frac{5 \cdot 14}{2 \cdot 14} + \frac{20}{7}$

Step 2.6.8

Multiply $\frac{20}{7}$ by $\frac{4}{4}$ .

$\frac{5 \cdot 7}{4 \cdot 7} + \frac{2 \cdot 28}{28} + \frac{5 \cdot 14}{2 \cdot 14} + \frac{20}{7} \cdot \frac{4}{4}$

Step 2.6.9

Multiply $\frac{20}{7}$ by $\frac{4}{4}$ .

$\frac{5 \cdot 7}{4 \cdot 7} + \frac{2 \cdot 28}{28} + \frac{5 \cdot 14}{2 \cdot 14} + \frac{20 \cdot 4}{7 \cdot 4}$

Step 2.6.10

Multiply $4$ by $7$ .

$\frac{5 \cdot 7}{28} + \frac{2 \cdot 28}{28} + \frac{5 \cdot 14}{2 \cdot 14} + \frac{20 \cdot 4}{7 \cdot 4}$

Step 2.6.11

Multiply $2$ by $14$ .

$\frac{5 \cdot 7}{28} + \frac{2 \cdot 28}{28} + \frac{5 \cdot 14}{28} + \frac{20 \cdot 4}{7 \cdot 4}$

Step 2.6.12

Reorder the factors of $7 \cdot 4$ .

$\frac{5 \cdot 7}{28} + \frac{2 \cdot 28}{28} + \frac{5 \cdot 14}{28} + \frac{20 \cdot 4}{4 \cdot 7}$

Step 2.6.13

Multiply $4$ by $7$ .

$\frac{5 \cdot 7}{28} + \frac{2 \cdot 28}{28} + \frac{5 \cdot 14}{28} + \frac{20 \cdot 4}{28}$

Step 2.7

Combine the numerators over the common denominator.

$\frac{5 \cdot 7 + 2 \cdot 28 + 5 \cdot 14 + 20 \cdot 4}{28}$

Step 2.8

Simplify each term.

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Step 2.8.1

Multiply $5$ by $7$ .

$\frac{35 + 2 \cdot 28 + 5 \cdot 14 + 20 \cdot 4}{28}$

Step 2.8.2

Multiply $2$ by $28$ .

$\frac{35 + 56 + 5 \cdot 14 + 20 \cdot 4}{28}$

Step 2.8.3

Multiply $5$ by $14$ .

$\frac{35 + 56 + 70 + 20 \cdot 4}{28}$

Step 2.8.4

Multiply $20$ by $4$ .

$\frac{35 + 56 + 70 + 80}{28}$

Step 2.9

Add $35$ and $56$ .

$\frac{91 + 70 + 80}{28}$

Step 2.10

Add $91$ and $70$ .

$\frac{161 + 80}{28}$

Step 2.11

Add $161$ and $80$ .

$\frac{241}{28}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{241}{28}$

Decimal Form:

$8.60714285 \dots$

Mixed Number Form:

$8 \frac{17}{28}$