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Chapter 0: Problem 118

Perform the indicated operations. $$\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right)$$

Short Answer

Expert verified
The final expression after performing all operations will be the result. The exact form will depend on how the algebraic simplification is carried out.

Step by step solution

01

Getting Rid of the Fractions

To make calculations easier, each fraction can be terminated. Simplify the expression inside each bracket by multiplying with the denominator of the fraction: \[ \left( \frac{x}{x} - \frac{1}{x} \right) \left( \frac{x+1}{x+1} - \frac{1}{x+1} \right) \left( \frac{x+2}{x+2} - \frac{1}{x+2} \right) \left( \frac{x+3}{x+3} - \frac{1}{x+3} \right) \] This will simplify to: \[ \left(1-\frac{1}{x}\right) \left(1-\frac{1}{x+1}\right) \left(1-\frac{1}{x+2}\right) \left(1-\frac{1}{x+3}\right) \]
02

Expanding the Brackets

Next expand the brackets, starting with the first and second bracket: \[ \left(1 - \frac{1}{x} - \frac{1}{x+1} + \frac{1}{x(x+1)}\right) \left(1-\frac{1}{x+2}\right) \left(1-\frac{1}{x+3}\right) \] Repeat this step for the third and fourth brackets as well.
03

Further Expansion and Simplification

The same approach is applied again, expanding and simplifying until all pairs of brackets have been dealt with, resulting in the final simplified expression.

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