Question

Find the value of $(1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \dots (1 - \frac{1}{n})$.

• A. $\frac{1}{n}$
• B. $\frac{1}{n+1}$
• C. $1 + \frac{1}{n}$
• D. $1 - \frac{1}{n}$

Hint:

For products or sums of series, always simplify the first few terms to identify a telescoping pattern where intermediate terms cancel out. This is a common technique for solving such problems.

Answer 1

The Correct Option is A

Step 1: Notice that each factor \(\left(1-\frac{1}{k}\right)\) simplifies to \(\frac{k-1}{k}\), so the whole product from \(k=2\) to \(k=n\) is \(\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\cdots\times\frac{n-1}{n}\).

Step 2: Define \(P(n)\) as the value of this product up to the term \(\frac{n-1}{n}\), and build it up one factor at a time: \(P(2)=\frac{1}{2}\), then \(P(3)=P(2)\times\frac{2}{3}=\frac{1}{2}\times\frac{2}{3}=\frac{1}{3}\), then \(P(4)=P(3)\times\frac{3}{4}=\frac{1}{3}\times\frac{3}{4}=\frac{1}{4}\).

Step 3: Each time we multiply by the next factor, the new numerator cancels with the previous result's denominator, so \(P(n)=\frac{1}{n}\) at every stage of this repeating pattern.
\[ \boxed{\frac{1}{n}} \]