Finite Math Examples

$1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1

Find the common denominator.

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Step 1.1

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

$\frac{1}{1} - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.2

Multiply $\frac{1}{1}$ by $\frac{2187}{2187}$ .

$\frac{1}{1} \cdot \frac{2187}{2187} - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.3

Multiply $\frac{1}{1}$ by $\frac{2187}{2187}$ .

$\frac{2187}{2187} - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.4

Multiply $\frac{1}{3}$ by $\frac{729}{729}$ .

$\frac{2187}{2187} - (\frac{1}{3} \cdot \frac{729}{729}) + \frac{1}{9} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.5

Multiply $\frac{1}{3}$ by $\frac{729}{729}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{1}{9} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.6

Multiply $\frac{1}{9}$ by $\frac{243}{243}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{1}{9} \cdot \frac{243}{243} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.7

Multiply $\frac{1}{9}$ by $\frac{243}{243}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - \frac{1}{27} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.8

Multiply $\frac{1}{27}$ by $\frac{81}{81}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - (\frac{1}{27} \cdot \frac{81}{81}) + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.9

Multiply $\frac{1}{27}$ by $\frac{81}{81}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - \frac{81}{27 \cdot 81} + \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.10

Multiply $\frac{1}{81}$ by $\frac{27}{27}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - \frac{81}{27 \cdot 81} + \frac{1}{81} \cdot \frac{27}{27} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.11

Multiply $\frac{1}{81}$ by $\frac{27}{27}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - \frac{81}{27 \cdot 81} + \frac{27}{81 \cdot 27} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187}$

Step 1.12

Multiply $\frac{1}{243}$ by $\frac{9}{9}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - \frac{81}{27 \cdot 81} + \frac{27}{81 \cdot 27} - (\frac{1}{243} \cdot \frac{9}{9}) + \frac{1}{729} - \frac{1}{2187}$

Step 1.13

Multiply $\frac{1}{243}$ by $\frac{9}{9}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - \frac{81}{27 \cdot 81} + \frac{27}{81 \cdot 27} - \frac{9}{243 \cdot 9} + \frac{1}{729} - \frac{1}{2187}$

Step 1.14

Multiply $\frac{1}{729}$ by $\frac{3}{3}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - \frac{81}{27 \cdot 81} + \frac{27}{81 \cdot 27} - \frac{9}{243 \cdot 9} + \frac{1}{729} \cdot \frac{3}{3} - \frac{1}{2187}$

Step 1.15

Multiply $\frac{1}{729}$ by $\frac{3}{3}$ .

$\frac{2187}{2187} - \frac{729}{3 \cdot 729} + \frac{243}{9 \cdot 243} - \frac{81}{27 \cdot 81} + \frac{27}{81 \cdot 27} - \frac{9}{243 \cdot 9} + \frac{3}{729 \cdot 3} - \frac{1}{2187}$

Step 1.16

Multiply $3$ by $729$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{9 \cdot 243} - \frac{81}{27 \cdot 81} + \frac{27}{81 \cdot 27} - \frac{9}{243 \cdot 9} + \frac{3}{729 \cdot 3} - \frac{1}{2187}$

Step 1.17

Multiply $9$ by $243$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{2187} - \frac{81}{27 \cdot 81} + \frac{27}{81 \cdot 27} - \frac{9}{243 \cdot 9} + \frac{3}{729 \cdot 3} - \frac{1}{2187}$

Step 1.18

Multiply $27$ by $81$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{2187} - \frac{81}{2187} + \frac{27}{81 \cdot 27} - \frac{9}{243 \cdot 9} + \frac{3}{729 \cdot 3} - \frac{1}{2187}$

Step 1.19

Reorder the factors of $81 \cdot 27$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{2187} - \frac{81}{2187} + \frac{27}{27 \cdot 81} - \frac{9}{243 \cdot 9} + \frac{3}{729 \cdot 3} - \frac{1}{2187}$

Step 1.20

Multiply $27$ by $81$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{2187} - \frac{81}{2187} + \frac{27}{2187} - \frac{9}{243 \cdot 9} + \frac{3}{729 \cdot 3} - \frac{1}{2187}$

Step 1.21

Reorder the factors of $243 \cdot 9$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{2187} - \frac{81}{2187} + \frac{27}{2187} - \frac{9}{9 \cdot 243} + \frac{3}{729 \cdot 3} - \frac{1}{2187}$

Step 1.22

Multiply $9$ by $243$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{2187} - \frac{81}{2187} + \frac{27}{2187} - \frac{9}{2187} + \frac{3}{729 \cdot 3} - \frac{1}{2187}$

Step 1.23

Reorder the factors of $729 \cdot 3$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{2187} - \frac{81}{2187} + \frac{27}{2187} - \frac{9}{2187} + \frac{3}{3 \cdot 729} - \frac{1}{2187}$

Step 1.24

Multiply $3$ by $729$ .

$\frac{2187}{2187} - \frac{729}{2187} + \frac{243}{2187} - \frac{81}{2187} + \frac{27}{2187} - \frac{9}{2187} + \frac{3}{2187} - \frac{1}{2187}$

Step 2

Combine the numerators over the common denominator.

$\frac{2187 - 729 + 243 - 81 + 27 - 9 + 3 - 1}{2187}$

Step 3

Simplify by adding and subtracting.

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Step 3.1

Subtract $729$ from $2187$ .

$\frac{1458 + 243 - 81 + 27 - 9 + 3 - 1}{2187}$

Step 3.2

Add $1458$ and $243$ .

$\frac{1701 - 81 + 27 - 9 + 3 - 1}{2187}$

Step 3.3

Subtract $81$ from $1701$ .

$\frac{1620 + 27 - 9 + 3 - 1}{2187}$

Step 3.4

Add $1620$ and $27$ .

$\frac{1647 - 9 + 3 - 1}{2187}$

Step 3.5

Subtract $9$ from $1647$ .

$\frac{1638 + 3 - 1}{2187}$

Step 3.6

Add $1638$ and $3$ .

$\frac{1641 - 1}{2187}$

Step 3.7

Subtract $1$ from $1641$ .

$\frac{1640}{2187}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1640}{2187}$

Decimal Form:

$0.74988568 \dots$