Solve 1/1*2+1/2*3+1/3+1/2019*2020

$$\frac{ 1 }{ 1 \cdot 2 } + \frac{ 1 }{ 2 \cdot 3 } + \frac{ 1 }{ 3 } + \frac{ 1 }{ 2019 \times 2020 }$$

Evaluate

$$\frac{4078381}{4078380}\approx 1.000000245$$

Solution Steps

Multiply 1 and 2 to get 2.

$$\frac{1}{2}+\frac{1}{2\times 3}+\frac{1}{3}+\frac{1}{2019\times 2020}$$

Multiply 2 and 3 to get 6.

$$\frac{1}{2}+\frac{1}{6}+\frac{1}{3}+\frac{1}{2019\times 2020}$$

Least common multiple of 2 and 6 is 6. Convert \frac{1}{2} and \frac{1}{6} to fractions with denominator 6.

$$\frac{3}{6}+\frac{1}{6}+\frac{1}{3}+\frac{1}{2019\times 2020}$$

Since \frac{3}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\frac{3+1}{6}+\frac{1}{3}+\frac{1}{2019\times 2020}$$

Add 3 and 1 to get 4.

$$\frac{4}{6}+\frac{1}{3}+\frac{1}{2019\times 2020}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$\frac{2}{3}+\frac{1}{3}+\frac{1}{2019\times 2020}$$

Since \frac{2}{3} and \frac{1}{3} have the same denominator, add them by adding their numerators.

$$\frac{2+1}{3}+\frac{1}{2019\times 2020}$$

Add 2 and 1 to get 3.

$$\frac{3}{3}+\frac{1}{2019\times 2020}$$

Divide 3 by 3 to get 1.

$$1+\frac{1}{2019\times 2020}$$

Multiply 2019 and 2020 to get 4078380.

$$1+\frac{1}{4078380}$$

Convert 1 to fraction \frac{4078380}{4078380}.

$$\frac{4078380}{4078380}+\frac{1}{4078380}$$

Since \frac{4078380}{4078380} and \frac{1}{4078380} have the same denominator, add them by adding their numerators.

$$\frac{4078380+1}{4078380}$$

Add 4078380 and 1 to get 4078381.

$$\frac{4078381}{4078380}$$

Factor

$$\frac{397 \cdot 10273}{2 ^ {2} \cdot 3 \cdot 5 \cdot 101 \cdot 673} = 1\frac{1}{4078380} = 1.0000002451953962$$