x(1-1%2F3%5E2)x(1-1%2F4%5E2).jpeg "(1-1/2^2)x(1-1/3^2)x(1-1/4^2)")
Understanding the Pattern: (1-1/2^2)x(1-1/3^2)x(1-1/4^2)...
This expression represents an interesting pattern that can be simplified and understood. Let's break it down step-by-step.
Identifying the Pattern
The expression follows a clear pattern:
- Each term is a product of two factors: (1 - 1/n^2) where 'n' is a consecutive integer starting from 2.
- The denominator of the fraction is the square of the integer 'n'.
Simplifying Each Term
We can simplify each term using the difference of squares factorization:
- (1 - 1/n^2) = (1 + 1/n)(1 - 1/n)
Expanding and Canceling Terms
Now let's expand the entire expression and see if we can cancel any terms:
- (1 + 1/2)(1 - 1/2) x (1 + 1/3)(1 - 1/3) x (1 + 1/4)(1 - 1/4) ...
Notice that each term has a pair of factors that cancel out:
- (1 + 1/2) is cancelled by (1 - 1/2) from the next term.
- (1 + 1/3) is cancelled by (1 - 1/3) from the next term.
- And so on...
The Final Result
This leaves us with only the first term:
- (1 + 1/2) = 3/2
Therefore, the expression (1-1/2^2)x(1-1/3^2)x(1-1/4^2)... simplifies to 3/2.
Generalization
This pattern can be generalized for any finite number of terms. If the expression ends with (1 - 1/n^2), the simplified result will be (n+1)/n.