(1-1/2^2)x(1-1/3^2)x(1-1/4^2)

3 min read Jun 16, 2024
(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.