x(1-1%2F3)x(1-1%2F4)x(1-1%2F5).jpeg "(1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)")
Unveiling the Pattern: (1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)
This mathematical expression might seem daunting at first glance, but it hides a simple and elegant pattern. Let's break it down step-by-step:
Simplifying the Expression
Each term in the expression takes the form of (1 - 1/n), where n is a consecutive integer starting from 2. We can simplify each term:
- (1 - 1/2) = 1/2
- (1 - 1/3) = 2/3
- (1 - 1/4) = 3/4
- (1 - 1/5) = 4/5
Now, our expression becomes: (1/2) x (2/3) x (3/4) x (4/5)
Recognizing the Cancellation
Notice that in the multiplication, many terms cancel out:
- The '2' in the numerator of the second term cancels with the '2' in the denominator of the first term.
- The '3' in the numerator of the third term cancels with the '3' in the denominator of the second term.
- And so on...
This leaves us with just the numerator of the last term and the denominator of the first term: 4/5.
The Solution
Therefore, the value of the expression (1-1/2)x(1-1/3)x(1-1/4)x(1-1/5) is 4/5.
Generalization
This pattern can be generalized for any number of terms. The expression:
(1 - 1/2) x (1 - 1/3) x (1 - 1/4) x ... x (1 - 1/n)
will always simplify to 1/n. This pattern demonstrates a beautiful connection between seemingly complex expressions and simple, elegant results.