Solve [1 + 1/(1 * 2) + 1/(1 * 2 * 4) + 1/(1 * 2 * 4 * 8) + 1/(1 * 2 * 4 * 8 * 16)] ) 16414 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$[ 1 + \frac { 1 } { 1 \times 2 } + \frac { 1 } { 1 \times 2 \times 4 } + \frac { 1 } { 1 \times 2 \times 4 \times 8 } + \frac { 1 } { 1 \times 2 \times 4 \times 8 \times 16 } ]$$

Evaluate

$\frac{1681}{1024}=1.641601562$

Solution Steps

Multiply $1$ and $2$ to get $2$.

$$1+\frac{1}{2}+\frac{1}{1\times 2\times 4}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Convert $1$ to fraction $\frac{2}{2}$.

$$\frac{2}{2}+\frac{1}{2}+\frac{1}{1\times 2\times 4}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Since $\frac{2}{2}$ and $\frac{1}{2}$ have the same denominator, add them by adding their numerators.

$$\frac{2+1}{2}+\frac{1}{1\times 2\times 4}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Add $2$ and $1$ to get $3$.

$$\frac{3}{2}+\frac{1}{1\times 2\times 4}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Multiply $1$ and $2$ to get $2$.

$$\frac{3}{2}+\frac{1}{2\times 4}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Multiply $2$ and $4$ to get $8$.

$$\frac{3}{2}+\frac{1}{8}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Least common multiple of $2$ and $8$ is $8$. Convert $\frac{3}{2}$ and $\frac{1}{8}$ to fractions with denominator $8$.

$$\frac{12}{8}+\frac{1}{8}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Since $\frac{12}{8}$ and $\frac{1}{8}$ have the same denominator, add them by adding their numerators.

$$\frac{12+1}{8}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Add $12$ and $1$ to get $13$.

$$\frac{13}{8}+\frac{1}{1\times 2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Multiply $1$ and $2$ to get $2$.

$$\frac{13}{8}+\frac{1}{2\times 4\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Multiply $2$ and $4$ to get $8$.

$$\frac{13}{8}+\frac{1}{8\times 8}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Multiply $8$ and $8$ to get $64$.

$$\frac{13}{8}+\frac{1}{64}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Least common multiple of $8$ and $64$ is $64$. Convert $\frac{13}{8}$ and $\frac{1}{64}$ to fractions with denominator $64$.

$$\frac{104}{64}+\frac{1}{64}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Since $\frac{104}{64}$ and $\frac{1}{64}$ have the same denominator, add them by adding their numerators.

$$\frac{104+1}{64}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Add $104$ and $1$ to get $105$.

$$\frac{105}{64}+\frac{1}{1\times 2\times 4\times 8\times 16}$$

Multiply $1$ and $2$ to get $2$.

$$\frac{105}{64}+\frac{1}{2\times 4\times 8\times 16}$$

Multiply $2$ and $4$ to get $8$.

$$\frac{105}{64}+\frac{1}{8\times 8\times 16}$$

Multiply $8$ and $8$ to get $64$.

$$\frac{105}{64}+\frac{1}{64\times 16}$$

Multiply $64$ and $16$ to get $1024$.

$$\frac{105}{64}+\frac{1}{1024}$$

Least common multiple of $64$ and $1024$ is $1024$. Convert $\frac{105}{64}$ and $\frac{1}{1024}$ to fractions with denominator $1024$.

$$\frac{1680}{1024}+\frac{1}{1024}$$

Since $\frac{1680}{1024}$ and $\frac{1}{1024}$ have the same denominator, add them by adding their numerators.

$$\frac{1680+1}{1024}$$

Add $1680$ and $1$ to get $1681$.

$$\frac{1681}{1024}$$

Factor

$\frac{41 ^ {2}}{2 ^ {10}} = 1\frac{657}{1024} = 1.6416015625$