x(1%2B1%2F3)x(1%2B1%2F4).jpeg "(1+1/2)x(1+1/3)x(1+1/4)")
Unlocking the Mystery of (1+1/2)x(1+1/3)x(1+1/4)
This expression might look intimidating at first glance, but it hides a simple and elegant solution. Let's break it down step by step:
Simplifying the Fractions
First, we need to simplify each term within the parentheses:
- 1 + 1/2 = 3/2
- 1 + 1/3 = 4/3
- 1 + 1/4 = 5/4
Now our expression looks like this: (3/2) x (4/3) x (5/4)
Multiplication of Fractions
Multiplying fractions is straightforward. We multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
(3/2) x (4/3) x (5/4) = (3 x 4 x 5) / (2 x 3 x 4)
Simplifying the Result
Before multiplying everything out, we can simplify by canceling out common factors in the numerator and denominator. In this case, we can cancel out the 3 and the 4:
(3 x 4 x 5) / (2 x 3 x 4) = 5/2
The Final Answer
Therefore, the solution to (1+1/2) x (1+1/3) x (1+1/4) is 5/2 or 2.5.
Hidden Pattern
Interestingly, this expression represents a pattern often seen in mathematics. It is related to the concept of telescoping series, where many terms cancel out, leaving only the first and last terms. If we continue this pattern, we'll find the general result for:
(1 + 1/2) x (1 + 1/3) x ... x (1 + 1/n) = (n+1)/2