Precalculus Examples

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1

Find the common denominator.

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Step 1.1

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

$\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.2

Multiply $\frac{1}{1}$ by $\frac{64}{64}$ .

$\frac{1}{1} \cdot \frac{64}{64} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.3

Multiply $\frac{1}{1}$ by $\frac{64}{64}$ .

$\frac{64}{64} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.4

Multiply $\frac{1}{2}$ by $\frac{32}{32}$ .

$\frac{64}{64} + \frac{1}{2} \cdot \frac{32}{32} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.5

Multiply $\frac{1}{2}$ by $\frac{32}{32}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.6

Multiply $\frac{1}{4}$ by $\frac{16}{16}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{1}{4} \cdot \frac{16}{16} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.7

Multiply $\frac{1}{4}$ by $\frac{16}{16}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{16}{4 \cdot 16} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.8

Multiply $\frac{1}{8}$ by $\frac{8}{8}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{16}{4 \cdot 16} + \frac{1}{8} \cdot \frac{8}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.9

Multiply $\frac{1}{8}$ by $\frac{8}{8}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{16}{4 \cdot 16} + \frac{8}{8 \cdot 8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1.10

Multiply $\frac{1}{16}$ by $\frac{4}{4}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{16}{4 \cdot 16} + \frac{8}{8 \cdot 8} + \frac{1}{16} \cdot \frac{4}{4} + \frac{1}{32} + \frac{1}{64}$

Step 1.11

Multiply $\frac{1}{16}$ by $\frac{4}{4}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{16}{4 \cdot 16} + \frac{8}{8 \cdot 8} + \frac{4}{16 \cdot 4} + \frac{1}{32} + \frac{1}{64}$

Step 1.12

Multiply $\frac{1}{32}$ by $\frac{2}{2}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{16}{4 \cdot 16} + \frac{8}{8 \cdot 8} + \frac{4}{16 \cdot 4} + \frac{1}{32} \cdot \frac{2}{2} + \frac{1}{64}$

Step 1.13

Multiply $\frac{1}{32}$ by $\frac{2}{2}$ .

$\frac{64}{64} + \frac{32}{2 \cdot 32} + \frac{16}{4 \cdot 16} + \frac{8}{8 \cdot 8} + \frac{4}{16 \cdot 4} + \frac{2}{32 \cdot 2} + \frac{1}{64}$

Step 1.14

Multiply $2$ by $32$ .

$\frac{64}{64} + \frac{32}{64} + \frac{16}{4 \cdot 16} + \frac{8}{8 \cdot 8} + \frac{4}{16 \cdot 4} + \frac{2}{32 \cdot 2} + \frac{1}{64}$

Step 1.15

Multiply $4$ by $16$ .

$\frac{64}{64} + \frac{32}{64} + \frac{16}{64} + \frac{8}{8 \cdot 8} + \frac{4}{16 \cdot 4} + \frac{2}{32 \cdot 2} + \frac{1}{64}$

Step 1.16

Multiply $8$ by $8$ .

$\frac{64}{64} + \frac{32}{64} + \frac{16}{64} + \frac{8}{64} + \frac{4}{16 \cdot 4} + \frac{2}{32 \cdot 2} + \frac{1}{64}$

Step 1.17

Reorder the factors of $16 \cdot 4$ .

$\frac{64}{64} + \frac{32}{64} + \frac{16}{64} + \frac{8}{64} + \frac{4}{4 \cdot 16} + \frac{2}{32 \cdot 2} + \frac{1}{64}$

Step 1.18

Multiply $4$ by $16$ .

$\frac{64}{64} + \frac{32}{64} + \frac{16}{64} + \frac{8}{64} + \frac{4}{64} + \frac{2}{32 \cdot 2} + \frac{1}{64}$

Step 1.19

Reorder the factors of $32 \cdot 2$ .

$\frac{64}{64} + \frac{32}{64} + \frac{16}{64} + \frac{8}{64} + \frac{4}{64} + \frac{2}{2 \cdot 32} + \frac{1}{64}$

Step 1.20

Multiply $2$ by $32$ .

$\frac{64}{64} + \frac{32}{64} + \frac{16}{64} + \frac{8}{64} + \frac{4}{64} + \frac{2}{64} + \frac{1}{64}$

Step 2

Combine the numerators over the common denominator.

$\frac{64 + 32 + 16 + 8 + 4 + 2 + 1}{64}$

Step 3

Simplify by adding numbers.

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Step 3.1

Add $64$ and $32$ .

$\frac{96 + 16 + 8 + 4 + 2 + 1}{64}$

Step 3.2

Add $96$ and $16$ .

$\frac{112 + 8 + 4 + 2 + 1}{64}$

Step 3.3

Add $112$ and $8$ .

$\frac{120 + 4 + 2 + 1}{64}$

Step 3.4

Add $120$ and $4$ .

$\frac{124 + 2 + 1}{64}$

Step 3.5

Add $124$ and $2$ .

$\frac{126 + 1}{64}$

Step 3.6

Add $126$ and $1$ .

$\frac{127}{64}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{127}{64}$

Decimal Form:

$1.984375$

Mixed Number Form:

$1 \frac{63}{64}$