Solve 2^2-1/2,3^2-1/3,4^2-1/4,5^2-1/5quad(12)-frac{1}{1*2},frac{1}{2*3},-frac{1}{3*4},frac{1}{4*5}

$$\frac { 2 ^ { 2 } - 1 } { 2 } , \frac { 3 ^ { 2 } - 1 } { 3 } , \frac { 4 ^ { 2 } - 1 } { 4 } , \frac { 5 ^ { 2 } - 1 } { 5 } \quad ( 12 ) - \frac { 1 } { 1 \times 2 } , \frac { 1 } { 2 \times 3 } , - \frac { 1 } { 3 \times 4 } , \frac { 1 } { 4 \times 5 }$$

Sort

$$-\frac{1}{12},\frac{1}{20},\frac{1}{6},\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{571}{10}$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

$$sort(\frac{4-1}{2},\frac{3^{2}-1}{3},\frac{4^{2}-1}{4},\frac{5^{2}-1}{5}\times 12-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Subtract 1 from 4 to get 3.

$$sort(\frac{3}{2},\frac{3^{2}-1}{3},\frac{4^{2}-1}{4},\frac{5^{2}-1}{5}\times 12-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Calculate 3 to the power of 2 and get 9.

$$sort(\frac{3}{2},\frac{9-1}{3},\frac{4^{2}-1}{4},\frac{5^{2}-1}{5}\times 12-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Subtract 1 from 9 to get 8.

$$sort(\frac{3}{2},\frac{8}{3},\frac{4^{2}-1}{4},\frac{5^{2}-1}{5}\times 12-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Calculate 4 to the power of 2 and get 16.

$$sort(\frac{3}{2},\frac{8}{3},\frac{16-1}{4},\frac{5^{2}-1}{5}\times 12-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Subtract 1 from 16 to get 15.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{5^{2}-1}{5}\times 12-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Calculate 5 to the power of 2 and get 25.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{25-1}{5}\times 12-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Subtract 1 from 25 to get 24.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{24}{5}\times 12-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Express \frac{24}{5}\times 12 as a single fraction.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{24\times 12}{5}-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Multiply 24 and 12 to get 288.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{288}{5}-\frac{1}{1\times 2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Multiply 1 and 2 to get 2.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{288}{5}-\frac{1}{2},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Least common multiple of 5 and 2 is 10. Convert \frac{288}{5} and \frac{1}{2} to fractions with denominator 10.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{576}{10}-\frac{5}{10},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Since \frac{576}{10} and \frac{5}{10} have the same denominator, subtract them by subtracting their numerators.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{576-5}{10},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Subtract 5 from 576 to get 571.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{571}{10},\frac{1}{2\times 3},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Multiply 2 and 3 to get 6.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{571}{10},\frac{1}{6},-\frac{1}{3\times 4},\frac{1}{4\times 5})$$

Multiply 3 and 4 to get 12.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{571}{10},\frac{1}{6},-\frac{1}{12},\frac{1}{4\times 5})$$

Multiply 4 and 5 to get 20.

$$sort(\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{571}{10},\frac{1}{6},-\frac{1}{12},\frac{1}{20})$$

Least common denominator of the numbers in the list \frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{571}{10},\frac{1}{6},-\frac{1}{12},\frac{1}{20} is 60. Convert numbers in the list to fractions with denominator 60.

$$\frac{90}{60},\frac{160}{60},\frac{225}{60},\frac{3426}{60},\frac{10}{60},-\frac{5}{60},\frac{3}{60}$$

To sort the list, start from a single element \frac{90}{60}.

$$\frac{90}{60}$$

Insert \frac{160}{60} to the appropriate location in the new list.

$$\frac{90}{60},\frac{160}{60}$$

Insert \frac{225}{60} to the appropriate location in the new list.

$$\frac{90}{60},\frac{160}{60},\frac{225}{60}$$

Insert \frac{3426}{60} to the appropriate location in the new list.

$$\frac{90}{60},\frac{160}{60},\frac{225}{60},\frac{3426}{60}$$

Insert \frac{10}{60} to the appropriate location in the new list.

$$\frac{10}{60},\frac{90}{60},\frac{160}{60},\frac{225}{60},\frac{3426}{60}$$

Insert -\frac{5}{60} to the appropriate location in the new list.

$$-\frac{5}{60},\frac{10}{60},\frac{90}{60},\frac{160}{60},\frac{225}{60},\frac{3426}{60}$$

Insert \frac{3}{60} to the appropriate location in the new list.

$$-\frac{5}{60},\frac{3}{60},\frac{10}{60},\frac{90}{60},\frac{160}{60},\frac{225}{60},\frac{3426}{60}$$

Replace the obtained fractions with the initial values.

$$-\frac{1}{12},\frac{1}{20},\frac{1}{6},\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{571}{10}$$

Evaluate

$$\frac{3}{2},\frac{8}{3},\frac{15}{4},\frac{571}{10},\frac{1}{6},-\frac{1}{12},\frac{1}{20}$$