Matemática discreta Ejemplos

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

Paso 1

Obtén el denominador común

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Paso 1.1

Escribe $1$ como una fracción con el denominador $1$ .

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

Paso 1.2

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

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

Paso 1.3

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

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

Paso 1.4

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

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

Paso 1.5

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

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

Paso 1.6

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

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

Paso 1.7

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

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

Paso 1.8

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

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

Paso 1.9

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

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

Paso 1.10

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

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

Paso 1.11

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

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

Paso 1.12

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

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

Paso 1.13

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

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

Paso 1.14

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

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

Paso 1.15

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

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

Paso 1.16

Multiplica $3$ por $729$ .

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

Paso 1.17

Multiplica $9$ por $243$ .

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

Paso 1.18

Multiplica $27$ por $81$ .

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

Paso 1.19

Reordena los factores de $81 \cdot 27$ .

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

Paso 1.20

Multiplica $27$ por $81$ .

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

Paso 1.21

Reordena los factores de $243 \cdot 9$ .

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

Paso 1.22

Multiplica $9$ por $243$ .

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

Paso 1.23

Reordena los factores de $729 \cdot 3$ .

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

Paso 1.24

Multiplica $3$ por $729$ .

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

Paso 2

Combina los numeradores sobre el denominador común.

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

Paso 3

Simplifica mediante suma y resta.

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Paso 3.1

Resta $729$ de $2187$ .

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

Paso 3.2

Suma $1458$ y $243$ .

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

Paso 3.3

Resta $81$ de $1701$ .

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

Paso 3.4

Suma $1620$ y $27$ .

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

Paso 3.5

Resta $9$ de $1647$ .

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

Paso 3.6

Suma $1638$ y $3$ .

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

Paso 3.7

Resta $1$ de $1641$ .

$\frac{1640}{2187}$

Paso 4

El resultado puede mostrarse de distintas formas.

Forma exacta:

$\frac{1640}{2187}$

Forma decimal:

$0.74988568 \dots$