Solve {l}{1/2+1/4+1/8+1/16+frac{1}{32}}{+frac{1}{64}+frac{1}{128}+frac{1}{258}}

$$\left. \begin{array} { l } { \frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } + \frac { 1 } { 32 } } \\ { + \frac { 1 } { 64 } + \frac { 1 } { 128 } + \frac { 1 } { 258 } } \end{array} \right.$$

Evaluate

$$\frac{16447}{16512}\approx 0.996063469$$

Solution Steps

Least common multiple of 2 and 4 is 4. Convert \frac{1}{2} and \frac{1}{4} to fractions with denominator 4.

$$\frac{2}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Since \frac{2}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{2+1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Add 2 and 1 to get 3.

$$\frac{3}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Least common multiple of 4 and 8 is 8. Convert \frac{3}{4} and \frac{1}{8} to fractions with denominator 8.

$$\frac{6}{8}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Since \frac{6}{8} and \frac{1}{8} have the same denominator, add them by adding their numerators.

$$\frac{6+1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Add 6 and 1 to get 7.

$$\frac{7}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Least common multiple of 8 and 16 is 16. Convert \frac{7}{8} and \frac{1}{16} to fractions with denominator 16.

$$\frac{14}{16}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Since \frac{14}{16} and \frac{1}{16} have the same denominator, add them by adding their numerators.

$$\frac{14+1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Add 14 and 1 to get 15.

$$\frac{15}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Least common multiple of 16 and 32 is 32. Convert \frac{15}{16} and \frac{1}{32} to fractions with denominator 32.

$$\frac{30}{32}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Since \frac{30}{32} and \frac{1}{32} have the same denominator, add them by adding their numerators.

$$\frac{30+1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Add 30 and 1 to get 31.

$$\frac{31}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Least common multiple of 32 and 64 is 64. Convert \frac{31}{32} and \frac{1}{64} to fractions with denominator 64.

$$\frac{62}{64}+\frac{1}{64}+\frac{1}{128}+\frac{1}{258}$$

Since \frac{62}{64} and \frac{1}{64} have the same denominator, add them by adding their numerators.

$$\frac{62+1}{64}+\frac{1}{128}+\frac{1}{258}$$

Add 62 and 1 to get 63.

$$\frac{63}{64}+\frac{1}{128}+\frac{1}{258}$$

Least common multiple of 64 and 128 is 128. Convert \frac{63}{64} and \frac{1}{128} to fractions with denominator 128.

$$\frac{126}{128}+\frac{1}{128}+\frac{1}{258}$$

Since \frac{126}{128} and \frac{1}{128} have the same denominator, add them by adding their numerators.

$$\frac{126+1}{128}+\frac{1}{258}$$

Add 126 and 1 to get 127.

$$\frac{127}{128}+\frac{1}{258}$$

Least common multiple of 128 and 258 is 16512. Convert \frac{127}{128} and \frac{1}{258} to fractions with denominator 16512.

$$\frac{16383}{16512}+\frac{64}{16512}$$

Since \frac{16383}{16512} and \frac{64}{16512} have the same denominator, add them by adding their numerators.

$$\frac{16383+64}{16512}$$

Add 16383 and 64 to get 16447.

$$\frac{16447}{16512}$$

Factor

$$\frac{16447}{2 ^ {7} \cdot 3 \cdot 43} = 0.9960634689922481$$