SOLUTION: Please help me solve this question. Find the value of n:(1-1/2)(2-2/3)(3-3/4)(4-4/5)...(15-15/16)=n!/16

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: Please help me solve this question. Find the value of n:(1-1/2)(2-2/3)(3-3/4)(4-4/5)...(15-15/16)=n!/16      Log On


   

Question 1020545: Please help me solve this question.
Find the value of n:(1-1/2)(2-2/3)(3-3/4)(4-4/5)...(15-15/16)=n!/16
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
%281-1%2F2%29%282-2%2F3%29%283-3%2F4%29...%2814-14%2F15%29%2815-15%2F16%29
Notice that the kth factor of this expression also has k as a factor,e.g., the factor 3-3/4 has 3 as a factor,and the 14th factor 14-14/15 has 14 as a factor.
If we factor all the k factors from each one of those factors, the expression becomes
1%2A2%2A3%2A4...13%2A14%2A15%281-1%2F2%29%281-1%2F3%29%281-1%2F4%29...%281-1%2F15%29%281-1%2F16%29
This expression is the same as
15%21%281%2F2%29%282%2F3%29%283%2F4%29...%2814%2F15%29%2815%2F16%29
The product of the fifteen fractional terms would just be equal to 1/16 by direct cancellation of common terms.
Hence the entire expression is equal to
15%21%281%2F16%29,
and so the value of n is highlight%2815%29.