SOLUTION: What is the value of the product (1-1/2^2)(1-1/3^2)(1-1/4^2)...(1-1/999^2)?

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: What is the value of the product (1-1/2^2)(1-1/3^2)(1-1/4^2)...(1-1/999^2)?      Log On


   

Question 1131575: What is the value of the product (1-1/2^2)(1-1/3^2)(1-1/4^2)...(1-1/999^2)?
Answer by ikleyn(52926) About Me  (Show Source):
You can put this solution on YOUR website!
.
First, it is clear that


    P = %281-1%2F2%5E2%29%281-1%2F3%5E2%29%281-1%2F4%5E2%29%2Aellipsis%2A%281-1%2F999%5E2%29 = 

         %281-1%2F2%29%2A%281-1%2F3%29%2A%281-1%2F4%29%2Aellipsis%2A%281-1%2F999%29*(%281%2B1%2F2%29%2A%281%2B1%2F3%29%2A%281%2B1%2F4%29%2Aellipsis%2A%281%2B1%2F999%29



Now, consider  Q = %281-1%2F2%29%2A%281-1%2F3%29%2A%281-1%2F4%29%2Aellipsis%2A%281-1%2F999%29.


    Notice that the denominator of the first fraction is the numerator of the second fraction.

    Next, the denominator of the second fraction is the numerator of the third fraction.

    This pattern continues: the denominator of the third fraction is the numerator of the fourth fraction.

    And so on . . . 


    Cancel all common factors in numerators and denominators to get the value  Q = 1%2F999.



Next, consider  R = %281%2B1%2F2%29%2A%281%2B1%2F3%29%2A%281%2B1%2F4%29%2Aellipsis%2A%281%2B1%2F999%29.


    Notice that the numerator of the first fraction is the denominator of the second fraction.

    Next, the numerator of the second fraction is the denominator of the third fraction.

    This pattern continues: the numerator of the third fraction is the denominator of the fourth fraction.

    And so on . . . 


    Cancel all common factors in numerators and denominators to get the value  R = %281%2F2%29%2A1000 = 500.



So, the final answer is  P = Q*R = %281%2F999%29%2A500 = 500%2F999.    ANSWER.

Solved.