x(1-1%2F3)x(1-1%2F4)x(1-1%2F5)x(1-1%2F6).jpeg "(1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6)")
Exploring a Pattern: (1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6)
This expression looks complicated at first glance, but it holds a hidden pattern that makes it easy to solve. Let's break it down:
Simplifying the Expression
Each term in the expression follows a simple pattern:
- (1 - 1/n) where 'n' is a consecutive integer starting from 2.
Let's simplify each term:
- (1 - 1/2) = 1/2
- (1 - 1/3) = 2/3
- (1 - 1/4) = 3/4
- (1 - 1/5) = 4/5
- (1 - 1/6) = 5/6
Now our expression becomes:
** (1/2) x (2/3) x (3/4) x (4/5) x (5/6) **
The Power of Cancellation
Notice something interesting? Every numerator after the first term cancels out with the denominator of the following term:
- 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.
- This pattern continues until we reach the last term.
This leaves us with:
1/6
Conclusion
The expression (1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6) simplifies to 1/6. The key to solving this lies in recognizing the pattern and utilizing the power of cancellation. This demonstrates a fundamental principle in mathematics - looking for patterns and simplifying expressions can often lead to elegant solutions.