FINd p and Q

Question

P = (1+1/2)(1+1/3)(1+1/4)..........(1+1/98)(1+1/99)

Q=(1-1/2)(1-1/3)...............................(1-1/99)(1-1/100)

P/Q = ?

Comments

5000...

Answer 1

P        = (1 + 1/2) * (1 + 1/3) * (1 + 1/4) * ...............(1 + 1/99)

          =  (3/2) * (4/3) * (5/4) * (6/5) .................(100/99)

So here barring the denominator of the 1st term and the numerator of the last term , all other numbers will be cancelled out as they appear exactly once each in numerator and denominator..

         =  100 / 2

         =   50

Now 

Q      =  (1 - 1/2) * (1 - 1/3) * (1 - 1/4) * .............. * (1 - 1/100)

         =  (1/2) * (2/3) * (3/4) * ............* (99/100)

Here barring numerator of first term and denominator of last term all others will be cancelled out..

        =   1/100

Thus  P/Q  =   50/[1/100]    =  5000 

Answer 2

Q = $\left ( 1- \frac{1}{2} \right ) \times \left ( 1- \frac{1}{3} \right )\times \left ( 1- \frac{1}{4} \right )..........\left ( 1- \frac{1}{100} \right )$

Q = $\left ( \frac{1}{2} \right ) \times \left (\frac{2}{3} \right )\times \left (\frac{3}{4} \right )..........\left ( \frac{99}{100} \right )$

Q = $\left ( \frac{1}{100} \right )$

 

P =  $\left ( 1+ \frac{1}{2} \right ) \times \left ( 1+ \frac{1}{3} \right )\times \left ( 1+ \frac{1}{4} \right )..........\left ( 1+ \frac{1}{99} \right)$

P = $\left ( \frac{3}{2} \right ) \times \left ( \frac{4}{3} \right )\times \left ( \frac{5}{4} \right )..........\left ( \frac{100}{99} \right)$

P = $\left ( \frac{100}{2} \right )$

P/Q = $\frac{\left ( \frac{100}{2} \right )}{\left ( \frac{1}{100} \right )}$ = 5000