x(1%2B1%2F3)x(1%2B1%2F4)x(1%2B1%2F5).jpeg "(1+1/2)x(1+1/3)x(1+1/4)x(1+1/5)")
Exploring the Pattern: (1+1/2)x(1+1/3)x(1+1/4)x(1+1/5)
This expression might seem simple at first glance, but it holds a fascinating pattern that leads to a surprising result. Let's delve into its breakdown and discover its hidden beauty.
Simplifying the Expression
First, we simplify each term in the expression:
- (1 + 1/2) = 3/2
- (1 + 1/3) = 4/3
- (1 + 1/4) = 5/4
- (1 + 1/5) = 6/5
Now, our expression becomes: (3/2) x (4/3) x (5/4) x (6/5)
Recognizing the Cancellation
Notice a beautiful pattern emerging! The numerator of each term cancels out with the denominator of the next term. This is known as telescoping.
- 3 in the numerator of the first term cancels with 3 in the denominator of the second term.
- 4 in the numerator of the second term cancels with 4 in the denominator of the third term.
- This pattern continues until we reach the final term.
The Final Result
After all the cancellations, we are left with:
(3/2) x (6/5) = 9/5
Therefore, the expression (1 + 1/2) x (1 + 1/3) x (1 + 1/4) x (1 + 1/5) simplifies to 9/5.
Conclusion
This seemingly straightforward expression reveals a remarkable pattern of cancellation. The concept of telescoping, where intermediate terms vanish, provides an elegant solution and emphasizes the power of recognizing hidden patterns in mathematics.