x(1-1%2F3)x(1-1%2F4)x(1-1%2F5)x(1-1%2F6)x(1-1%2F7)x(1-1%2F8)x(1-1%2F9).jpeg "(1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6)x(1-1/7)x(1-1/8)x(1-1/9)")
Unveiling the Pattern: A Journey Through Multiplicative Simplification
The expression (1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6)x(1-1/7)x(1-1/8)x(1-1/9) appears complex at first glance, but beneath its surface lies a fascinating pattern that simplifies the calculation. Let's embark on a journey to unravel this mathematical puzzle.
Simplifying the Expression
Each term within the expression can be rewritten as a fraction:
- (1-1/2) = 1/2
- (1-1/3) = 2/3
- (1-1/4) = 3/4
- ... and so on.
Therefore, our original expression becomes:
(1/2) x (2/3) x (3/4) x (4/5) x (5/6) x (6/7) x (7/8) x (8/9)
The Art of Cancellation
Notice that a beautiful pattern emerges. The numerator of each fraction cancels out with the denominator of the subsequent fraction. This "telescoping" effect significantly simplifies our calculation:
- (1/2) x (2 / 3) x (3 / 4) x (4 / 5) x (5 / 6) x (6 / 7) x (7 / 8) x (8 / 9)
We are left with only the first numerator and the last denominator:
1/9
Conclusion
Through a series of simple transformations and a neat cancellation pattern, we've determined that (1-1/2)x(1-1/3)x(1-1/4)x(1-1/5)x(1-1/6)x(1-1/7)x(1-1/8)x(1-1/9) is equal to 1/9. This example highlights the power of pattern recognition in simplifying seemingly complex mathematical expressions.