SOLUTION: Determine the value of the product (1- 1/2^2)(1-1/3^2)(1-1/4^2)....(1-1/99^2)(1-1/100^2)

Question 252454: Determine the value of the product
(1- 1/2^2)(1-1/3^2)(1-1/4^2)....(1-1/99^2)(1-1/100^2)
Answer by drk(1908) (Show Source): You can put this solution on YOUR website!
We want the product of
(1-1/2^2)(1-1/3^2)(1-1/4^2)....(1-1/99^2)(1-1/100^2).
We can re-express this a the product of simple fractions as
(3/4)(8/9)(15/16)(24/25) . . . (9999/10000).
(3/4)(8/9) = 2/3 = [6/9]
(3/4)(8/9)(15/16) = 5/8 = [10/16]
(3/4)(8/9)(15/16)(24/25) = 3/5 = [15/25]
(3/4)(8/9)(15/16)(24/25)(35/36) = 7/12 = [21/36]
. . .
Notice the [parentheses] fractions.
There is a pattern to the numerator: 6, 10, 15, 21, . . .are all triangular numbers with the formula
where n>=2.
The denominator is just (n+1)^2, where n>=2.
Together, we get
/ , n>=2.
I use two because the first multiplication took 2 fractions.
Now all of these together are 99 terms, so n = 99.
Using n = 99, we get
/ ~ .505

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