The expression $\left(1 + \frac{1}{3}\right)\left(1 + \frac{1}{4}\right)\left(1 + \frac{1}{5}\right)\cdots\left(1 + \frac{1}{n}\right)$ simplifies to

### Concept & Formula This represents an additive telescoping product. When terms are simplified, the numerator of each fraction cancels out with the denominator of the subsequent fraction. ### Step-by-Step Solution Simplify each term in the product: $$1 + \frac{1}{3} = \frac{4}{3}$$ $$1 + \frac{1}{4} = \frac{5}{4}$$ $$1 + \frac{1}{5} = \frac{6}{5}$$ $$\vdots$$ $$1 + \frac{1}{n} = \frac{n + 1}{n}$$ Write the complete product using these simplified forms: $$\left(\frac{4}{3}\right) \times \left(\frac{5}{4}\right) \times \left(\frac{6}{5}\right) \times \cdots \times \left(\frac{n}{n-1}\right) \times \left(\frac{n+1}{n}\right)$$ Observe the cancellation pattern: the numerator of the first term ($4$) cancels with the denominator of the second term ($4$), and so on. The only terms that survive the cancellation are the first denominator and the last numerator: $$\frac{n + 1}{3}$$ ### Exam Strategy & Shortcut Test the series with a small value, say $n = 4$: $$\left(1 + \frac{1}{3}\right)\left(1 + \frac{1}{4}\right) = \frac{4}{3} \times \frac{5}{4} = \frac{5}{3}$$ Substitute $n = 4$ into the options: Option (a): $\frac{4 + 1}{3} = \frac{5}{3}$ This perfectly matches our tested result. ### Common Pitfall Ensure you track whether it is the numerator or the denominator that survives at each end. For positive terms ($1 + \frac{1}{k}$), the larger value survives on top, and the smaller value survives on the bottom. ### Final Answer **Therefore, the correct answer is (n + 1)/3.**