The simplest value of {(1+1/2)(1+1/3)(1+1/4)......(1+1/50)} / {(1-1/2)(1-1/3)(1-1/4)....(1-1/50)} is

Question

The simplest value of \(\cfrac{\left(1+\frac12\right)\left(1+\frac13\right)\left(1+\frac14\right)....\left(1+\frac{1}{50}\right)}{\left(1-\frac12\right)\left(1-\frac13\right)\left(1-\frac14\right)....\left(1-\frac{1}{50}\right)}\) is

(a) \(\frac{7}{32}\)

(b) \(\frac47\)

(c) 1275

(d) \(\frac{256}{343}\)

Answer 1

Correct option : (c) 1275

\(\cfrac{\left(1+\frac12\right)\left(1+\frac13\right)\left(1+\frac14\right)....\left(1+\frac{1}{50}\right)}{\left(1-\frac12\right)\left(1-\frac13\right)\left(1-\frac14\right)....\left(1-\frac{1}{50}\right)}\)

\(=\cfrac{\cfrac32\times\cfrac43\times \cfrac54\times....\times\cfrac{51}{50}}{\cfrac12\times\cfrac23\times\cfrac34\times....\times\cfrac{49}{50}}\)

\(=\frac{\cfrac{51}{2}}{\cfrac{1}{50}} = \cfrac{51}{2}\times 50 =51\times25=1275.\)