The value of $[1 + \frac{1}{x+1}][1 + \frac{1}{x+2}][1 + \frac{1}{x+3}][1 + \frac{1}{x+4}]$ is

### Concept & Formula This problem utilizes a mathematical pattern known as a "telescoping product." By simplifying each bracketed fraction individually by finding a common denominator, you will reveal a chain of fractions where the numerator of one term perfectly cancels out the denominator of the subsequent term. $$ \text{Simplification step: } 1 + \frac{a}{b} = \frac{b + a}{b} $$ ### Step-by-Step Solution * **Given:** * Expression: $[1 + \frac{1}{x+1}][1 + \frac{1}{x+2}][1 + \frac{1}{x+3}][1 + \frac{1}{x+4}]$ * **Calculation:** 1. Simplify the first bracket by taking $(x+1)$ as the common denominator: $1 + \frac{1}{x+1} = \frac{x+1+1}{x+1} = \frac{x+2}{x+1}$ 2. Simplify the second bracket using the same method: $1 + \frac{1}{x+2} = \frac{x+2+1}{x+2} = \frac{x+3}{x+2}$ 3. Simplify the third bracket: $1 + \frac{1}{x+3} = \frac{x+3+1}{x+3} = \frac{x+4}{x+3}$ 4. Simplify the fourth bracket: $1 + \frac{1}{x+4} = \frac{x+4+1}{x+4} = \frac{x+5}{x+4}$ 5. Rewrite the entire original expression by multiplying these simplified fractions together: $[\frac{x+2}{x+1}] \cdot [\frac{x+3}{x+2}] \cdot [\frac{x+4}{x+3}] \cdot [\frac{x+5}{x+4}]$ 6. Observe the pattern: The numerator $(x+2)$ in the first term cancels the denominator $(x+2)$ in the second term. The $(x+3)$ terms cancel, and the $(x+4)$ terms cancel. 7. After all cross-cancellations, you are left with only the final numerator and the very first denominator: $\frac{x+5}{x+1}$. ### Exam Strategy & Shortcut Whenever you see a sequential product of fractions formatted as $(1 + \frac{1}{n})$, immediately recognize it as a telescoping series. You do not need to write out every step. Mentally evaluate the first term to find the starting denominator, and evaluate the last term to find the ending numerator. The answer will always be $\frac{\text{Final Numerator}}{\text{First Denominator}}$. ### Common Pitfall The primary mistake is failing to identify the pattern and attempting to multiply the complex binomials directly (e.g., trying to use FOIL on the fractions). This leads to a massive polynomial expression that is virtually impossible to untangle and consumes excessive exam time. ### Final Answer Therefore, the correct answer is $\frac{x+5}{x+1}$.