Solve 1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+frac{1}{(x+4)(x+5)}

$$\frac { 1 } { x ( x + 1 ) } + \frac { 1 } { ( x + 1 ) ( x + 2 ) } + \frac { 1 } { ( x + 2 ) ( x + 3 ) } + \frac { 1 } { ( x + 3 ) ( x + 4 ) } + \frac { 1 } { ( x + 4 ) ( x + 5 ) }$$

Evaluate

$$\frac{5}{x\left(x+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and \left(x+1\right)\left(x+2\right) is x\left(x+1\right)\left(x+2\right). Multiply \frac{1}{x\left(x+1\right)} times \frac{x+2}{x+2}. Multiply \frac{1}{\left(x+1\right)\left(x+2\right)} times \frac{x}{x}.

$$\frac{x+2}{x\left(x+1\right)\left(x+2\right)}+\frac{x}{x\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Since \frac{x+2}{x\left(x+1\right)\left(x+2\right)} and \frac{x}{x\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+2+x}{x\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Combine like terms in x+2+x.

$$\frac{2x+2}{x\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{2x+2}{x\left(x+1\right)\left(x+2\right)}.

$$\frac{2\left(x+1\right)}{x\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{2}{x\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and \left(x+2\right)\left(x+3\right) is x\left(x+2\right)\left(x+3\right). Multiply \frac{2}{x\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{1}{\left(x+2\right)\left(x+3\right)} times \frac{x}{x}.

$$\frac{2\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)}+\frac{x}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Since \frac{2\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)} and \frac{x}{x\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(x+3\right)+x}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Do the multiplications in 2\left(x+3\right)+x.

$$\frac{2x+6+x}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Combine like terms in 2x+6+x.

$$\frac{3x+6}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{3x+6}{x\left(x+2\right)\left(x+3\right)}.

$$\frac{3\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{3}{x\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+3\right) and \left(x+3\right)\left(x+4\right) is x\left(x+3\right)\left(x+4\right). Multiply \frac{3}{x\left(x+3\right)} times \frac{x+4}{x+4}. Multiply \frac{1}{\left(x+3\right)\left(x+4\right)} times \frac{x}{x}.

$$\frac{3\left(x+4\right)}{x\left(x+3\right)\left(x+4\right)}+\frac{x}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Since \frac{3\left(x+4\right)}{x\left(x+3\right)\left(x+4\right)} and \frac{x}{x\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(x+4\right)+x}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Do the multiplications in 3\left(x+4\right)+x.

$$\frac{3x+12+x}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Combine like terms in 3x+12+x.

$$\frac{4x+12}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{4x+12}{x\left(x+3\right)\left(x+4\right)}.

$$\frac{4\left(x+3\right)}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{4}{x\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+4\right) and \left(x+4\right)\left(x+5\right) is x\left(x+4\right)\left(x+5\right). Multiply \frac{4}{x\left(x+4\right)} times \frac{x+5}{x+5}. Multiply \frac{1}{\left(x+4\right)\left(x+5\right)} times \frac{x}{x}.

$$\frac{4\left(x+5\right)}{x\left(x+4\right)\left(x+5\right)}+\frac{x}{x\left(x+4\right)\left(x+5\right)}$$

Since \frac{4\left(x+5\right)}{x\left(x+4\right)\left(x+5\right)} and \frac{x}{x\left(x+4\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x+5\right)+x}{x\left(x+4\right)\left(x+5\right)}$$

Do the multiplications in 4\left(x+5\right)+x.

$$\frac{4x+20+x}{x\left(x+4\right)\left(x+5\right)}$$

Combine like terms in 4x+20+x.

$$\frac{5x+20}{x\left(x+4\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{5x+20}{x\left(x+4\right)\left(x+5\right)}.

$$\frac{5\left(x+4\right)}{x\left(x+4\right)\left(x+5\right)}$$

Cancel out x+4 in both numerator and denominator.

$$\frac{5}{x\left(x+5\right)}$$

Expand x\left(x+5\right).

$$\frac{5}{x^{2}+5x}$$

Expand

$$\frac{5}{x\left(x+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and \left(x+1\right)\left(x+2\right) is x\left(x+1\right)\left(x+2\right). Multiply \frac{1}{x\left(x+1\right)} times \frac{x+2}{x+2}. Multiply \frac{1}{\left(x+1\right)\left(x+2\right)} times \frac{x}{x}.

$$\frac{x+2}{x\left(x+1\right)\left(x+2\right)}+\frac{x}{x\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Since \frac{x+2}{x\left(x+1\right)\left(x+2\right)} and \frac{x}{x\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+2+x}{x\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Combine like terms in x+2+x.

$$\frac{2x+2}{x\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{2x+2}{x\left(x+1\right)\left(x+2\right)}.

$$\frac{2\left(x+1\right)}{x\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{2}{x\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+2\right) and \left(x+2\right)\left(x+3\right) is x\left(x+2\right)\left(x+3\right). Multiply \frac{2}{x\left(x+2\right)} times \frac{x+3}{x+3}. Multiply \frac{1}{\left(x+2\right)\left(x+3\right)} times \frac{x}{x}.

$$\frac{2\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)}+\frac{x}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Since \frac{2\left(x+3\right)}{x\left(x+2\right)\left(x+3\right)} and \frac{x}{x\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(x+3\right)+x}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Do the multiplications in 2\left(x+3\right)+x.

$$\frac{2x+6+x}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Combine like terms in 2x+6+x.

$$\frac{3x+6}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{3x+6}{x\left(x+2\right)\left(x+3\right)}.

$$\frac{3\left(x+2\right)}{x\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{3}{x\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+3\right) and \left(x+3\right)\left(x+4\right) is x\left(x+3\right)\left(x+4\right). Multiply \frac{3}{x\left(x+3\right)} times \frac{x+4}{x+4}. Multiply \frac{1}{\left(x+3\right)\left(x+4\right)} times \frac{x}{x}.

$$\frac{3\left(x+4\right)}{x\left(x+3\right)\left(x+4\right)}+\frac{x}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Since \frac{3\left(x+4\right)}{x\left(x+3\right)\left(x+4\right)} and \frac{x}{x\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(x+4\right)+x}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Do the multiplications in 3\left(x+4\right)+x.

$$\frac{3x+12+x}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Combine like terms in 3x+12+x.

$$\frac{4x+12}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{4x+12}{x\left(x+3\right)\left(x+4\right)}.

$$\frac{4\left(x+3\right)}{x\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{4}{x\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+4\right) and \left(x+4\right)\left(x+5\right) is x\left(x+4\right)\left(x+5\right). Multiply \frac{4}{x\left(x+4\right)} times \frac{x+5}{x+5}. Multiply \frac{1}{\left(x+4\right)\left(x+5\right)} times \frac{x}{x}.

$$\frac{4\left(x+5\right)}{x\left(x+4\right)\left(x+5\right)}+\frac{x}{x\left(x+4\right)\left(x+5\right)}$$

Since \frac{4\left(x+5\right)}{x\left(x+4\right)\left(x+5\right)} and \frac{x}{x\left(x+4\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x+5\right)+x}{x\left(x+4\right)\left(x+5\right)}$$

Do the multiplications in 4\left(x+5\right)+x.

$$\frac{4x+20+x}{x\left(x+4\right)\left(x+5\right)}$$

Combine like terms in 4x+20+x.

$$\frac{5x+20}{x\left(x+4\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{5x+20}{x\left(x+4\right)\left(x+5\right)}.

$$\frac{5\left(x+4\right)}{x\left(x+4\right)\left(x+5\right)}$$

Cancel out x+4 in both numerator and denominator.

$$\frac{5}{x\left(x+5\right)}$$

Expand x\left(x+5\right).

$$\frac{5}{x^{2}+5x}$$