The Amazing Pattern of Fractions
Have you ever noticed a fascinating pattern in fractions? Let's explore an intriguing sequence of fractions that will leave you amazed.
The Pattern
The sequence starts with:
1 1/2 × 1 1/3 × 1 1/4 × 1 1/5 × ... × 1 1/2005 × 1 1/2006 × 1 1/2007
At first glance, it might seem like a complex and daunting task to calculate the product of these fractions. However, as we delve deeper, you'll be surprised to find a remarkable property that emerges.
The Magic of Cancellation
As we multiply the fractions, something incredible happens. The numerators (the numbers on top) and the denominators (the numbers at the bottom) start to cancel each other out.
Let's illustrate this with the first few fractions:
(1 1/2) × (1 1/3) = (3/2) × (4/3) = 12/6 = 2
(2) × (1 1/4) = 2 × (5/4) = 10/4 = 2.5
(2.5) × (1 1/5) = 2.5 × (6/5) = 15/5 = 3
Do you see the pattern emerging? The product of the fractions is always a whole number!
The Astonishing Result
As we continue multiplying the fractions, the numerators and denominators will continue to cancel each other out, ultimately resulting in a staggering product:
1 1/2 × 1 1/3 × 1 1/4 × 1 1/5 × ... × 1 1/2005 × 1 1/2006 × 1 1/2007 = 2007
Yes, you read that correctly! The product of these fractions is the last numerator, which is 2007.
Conclusion
This incredible pattern of fractions is a testament to the beauty and simplicity of mathematics. It's a reminder that even the most complex-looking problems can have surprisingly elegant solutions. So, the next time you encounter a daunting mathematical problem, take a step back, and look for the hidden patterns waiting to be uncovered.
