x(1%2B1%2F3)x(1%2B1%2F4)x(1%2B1%2Fn).jpeg "(1+1/2)x(1+1/3)x(1+1/4)x(1+1/n)")
Exploring the Pattern: (1+1/2)x(1+1/3)x(1+1/4)x(1+1/n)
This intriguing sequence, (1+1/2)x(1+1/3)x(1+1/4)x(1+1/n), holds a fascinating mathematical pattern. Let's dive into understanding its behavior and uncovering its simplified form.
Expanding the Sequence
First, let's expand the initial terms to get a better grasp of the pattern:
- (1 + 1/2) = 3/2
- (1 + 1/3) = 4/3
- (1 + 1/4) = 5/4
Notice how the numerators increase by one, while the denominators follow the natural numbers. This pattern continues for each subsequent term:
- (1 + 1/n) = (n+1)/n
The Product of the Terms
Now, let's consider the product of the first few terms:
(3/2) x (4/3) x (5/4) = 15/4
Observe that the numerators and denominators simplify:
- The denominator of each term cancels with the numerator of the next term.
- We are left with the numerator of the last term (n+1) and the denominator of the first term (2).
The General Form
Based on the pattern observed, we can generalize the product for any positive integer 'n':
(1+1/2)x(1+1/3)x(1+1/4)x....x(1+1/n) = (n+1)/2
Conclusion
This sequence demonstrates a beautiful example of how seemingly complex patterns can be simplified through careful observation and generalization. Understanding this pattern allows us to calculate the product of the series up to any given 'n' without performing tedious multiplication.