Solve 1/1*3+1/3*5+1/5*7+1/7*9+frac{1}{8*11}+frac{1}{11*13}+frac{1}{13*15}+frac{1}{15*17}+frac{1}{1749}

$$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \frac { 1 } { 7 \times 9 } + \frac { 1 } { 8 \times 11 } + \frac { 1 } { 11 \times 13 } + \frac { 1 } { 13 \times 15 } + \frac { 1 } { 15 \times 17 } + \frac { 1 } { 1749 }$$

Evaluate

$$\frac{30647}{64872}\approx 0.472422617$$

Solution Steps

Multiply 1 and 3 to get 3.

$$\frac{1}{3}+\frac{1}{3\times 5}+\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Multiply 3 and 5 to get 15.

$$\frac{1}{3}+\frac{1}{15}+\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Least common multiple of 3 and 15 is 15. Convert \frac{1}{3} and \frac{1}{15} to fractions with denominator 15.

$$\frac{5}{15}+\frac{1}{15}+\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Since \frac{5}{15} and \frac{1}{15} have the same denominator, add them by adding their numerators.

$$\frac{5+1}{15}+\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Add 5 and 1 to get 6.

$$\frac{6}{15}+\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Reduce the fraction \frac{6}{15} to lowest terms by extracting and canceling out 3.

$$\frac{2}{5}+\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Multiply 5 and 7 to get 35.

$$\frac{2}{5}+\frac{1}{35}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Least common multiple of 5 and 35 is 35. Convert \frac{2}{5} and \frac{1}{35} to fractions with denominator 35.

$$\frac{14}{35}+\frac{1}{35}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Since \frac{14}{35} and \frac{1}{35} have the same denominator, add them by adding their numerators.

$$\frac{14+1}{35}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Add 14 and 1 to get 15.

$$\frac{15}{35}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Reduce the fraction \frac{15}{35} to lowest terms by extracting and canceling out 5.

$$\frac{3}{7}+\frac{1}{7\times 9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Multiply 7 and 9 to get 63.

$$\frac{3}{7}+\frac{1}{63}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Least common multiple of 7 and 63 is 63. Convert \frac{3}{7} and \frac{1}{63} to fractions with denominator 63.

$$\frac{27}{63}+\frac{1}{63}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Since \frac{27}{63} and \frac{1}{63} have the same denominator, add them by adding their numerators.

$$\frac{27+1}{63}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Add 27 and 1 to get 28.

$$\frac{28}{63}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Reduce the fraction \frac{28}{63} to lowest terms by extracting and canceling out 7.

$$\frac{4}{9}+\frac{1}{8\times 11}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Multiply 8 and 11 to get 88.

$$\frac{4}{9}+\frac{1}{88}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Least common multiple of 9 and 88 is 792. Convert \frac{4}{9} and \frac{1}{88} to fractions with denominator 792.

$$\frac{352}{792}+\frac{9}{792}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Since \frac{352}{792} and \frac{9}{792} have the same denominator, add them by adding their numerators.

$$\frac{352+9}{792}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Add 352 and 9 to get 361.

$$\frac{361}{792}+\frac{1}{11\times 13}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Multiply 11 and 13 to get 143.

$$\frac{361}{792}+\frac{1}{143}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Least common multiple of 792 and 143 is 10296. Convert \frac{361}{792} and \frac{1}{143} to fractions with denominator 10296.

$$\frac{4693}{10296}+\frac{72}{10296}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Since \frac{4693}{10296} and \frac{72}{10296} have the same denominator, add them by adding their numerators.

$$\frac{4693+72}{10296}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Add 4693 and 72 to get 4765.

$$\frac{4765}{10296}+\frac{1}{13\times 15}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Multiply 13 and 15 to get 195.

$$\frac{4765}{10296}+\frac{1}{195}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Least common multiple of 10296 and 195 is 51480. Convert \frac{4765}{10296} and \frac{1}{195} to fractions with denominator 51480.

$$\frac{23825}{51480}+\frac{264}{51480}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Since \frac{23825}{51480} and \frac{264}{51480} have the same denominator, add them by adding their numerators.

$$\frac{23825+264}{51480}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Add 23825 and 264 to get 24089.

$$\frac{24089}{51480}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Reduce the fraction \frac{24089}{51480} to lowest terms by extracting and canceling out 13.

$$\frac{1853}{3960}+\frac{1}{15\times 17}+\frac{1}{1749}$$

Multiply 15 and 17 to get 255.

$$\frac{1853}{3960}+\frac{1}{255}+\frac{1}{1749}$$

Least common multiple of 3960 and 255 is 67320. Convert \frac{1853}{3960} and \frac{1}{255} to fractions with denominator 67320.

$$\frac{31501}{67320}+\frac{264}{67320}+\frac{1}{1749}$$

Since \frac{31501}{67320} and \frac{264}{67320} have the same denominator, add them by adding their numerators.

$$\frac{31501+264}{67320}+\frac{1}{1749}$$

Add 31501 and 264 to get 31765.

$$\frac{31765}{67320}+\frac{1}{1749}$$

Reduce the fraction \frac{31765}{67320} to lowest terms by extracting and canceling out 5.

$$\frac{6353}{13464}+\frac{1}{1749}$$

Least common multiple of 13464 and 1749 is 713592. Convert \frac{6353}{13464} and \frac{1}{1749} to fractions with denominator 713592.

$$\frac{336709}{713592}+\frac{408}{713592}$$

Since \frac{336709}{713592} and \frac{408}{713592} have the same denominator, add them by adding their numerators.

$$\frac{336709+408}{713592}$$

Add 336709 and 408 to get 337117.

$$\frac{337117}{713592}$$

Reduce the fraction \frac{337117}{713592} to lowest terms by extracting and canceling out 11.

$$\frac{30647}{64872}$$

Factor

$$\frac{19 \cdot 1613}{2 ^ {3} \cdot 3 ^ {2} \cdot 17 \cdot 53} = 0.4724226168454803$$