Matemática discreta Exemplos

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

Etapa 1

Encontre o denominador comum.

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Etapa 1.1

Escreva $1$ como uma fração com denominador $1$ .

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

Etapa 1.2

Multiplique $\frac{1}{1}$ por $\frac{2187}{2187}$ .

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

Etapa 1.3

Multiplique $\frac{1}{1}$ por $\frac{2187}{2187}$ .

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

Etapa 1.4

Multiplique $\frac{1}{3}$ por $\frac{729}{729}$ .

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

Etapa 1.5

Multiplique $\frac{1}{3}$ por $\frac{729}{729}$ .

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

Etapa 1.6

Multiplique $\frac{1}{9}$ por $\frac{243}{243}$ .

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

Etapa 1.7

Multiplique $\frac{1}{9}$ por $\frac{243}{243}$ .

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

Etapa 1.8

Multiplique $\frac{1}{27}$ por $\frac{81}{81}$ .

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

Etapa 1.9

Multiplique $\frac{1}{27}$ por $\frac{81}{81}$ .

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

Etapa 1.10

Multiplique $\frac{1}{81}$ por $\frac{27}{27}$ .

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

Etapa 1.11

Multiplique $\frac{1}{81}$ por $\frac{27}{27}$ .

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

Etapa 1.12

Multiplique $\frac{1}{243}$ por $\frac{9}{9}$ .

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

Etapa 1.13

Multiplique $\frac{1}{243}$ por $\frac{9}{9}$ .

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

Etapa 1.14

Multiplique $\frac{1}{729}$ por $\frac{3}{3}$ .

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

Etapa 1.15

Multiplique $\frac{1}{729}$ por $\frac{3}{3}$ .

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

Etapa 1.16

Multiplique $3$ por $729$ .

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

Etapa 1.17

Multiplique $9$ por $243$ .

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

Etapa 1.18

Multiplique $27$ por $81$ .

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

Etapa 1.19

Reordene os fatores de $81 \cdot 27$ .

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

Etapa 1.20

Multiplique $27$ por $81$ .

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

Etapa 1.21

Reordene os fatores de $243 \cdot 9$ .

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

Etapa 1.22

Multiplique $9$ por $243$ .

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

Etapa 1.23

Reordene os fatores de $729 \cdot 3$ .

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

Etapa 1.24

Multiplique $3$ por $729$ .

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

Etapa 2

Combine os numeradores em relação ao denominador comum.

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

Etapa 3

Simplifique somando e subtraindo.

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Etapa 3.1

Subtraia $729$ de $2187$ .

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

Etapa 3.2

Some $1458$ e $243$ .

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

Etapa 3.3

Subtraia $81$ de $1701$ .

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

Etapa 3.4

Some $1620$ e $27$ .

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

Etapa 3.5

Subtraia $9$ de $1647$ .

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

Etapa 3.6

Some $1638$ e $3$ .

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

Etapa 3.7

Subtraia $1$ de $1641$ .

$\frac{1640}{2187}$

Etapa 4

O resultado pode ser mostrado de várias formas.

Forma exata:

$\frac{1640}{2187}$

Forma decimal:

$0.74988568 \dots$