option c

Solution:

Let first two numbers be x and y.. so the sequence will be

$x,y,\frac{y}{x},\frac{1}{x},\frac{1}{y},\frac{x}{y},x,y,..$ the sequence repeats after 6 numbers. Also the product of first six numbers is 1.

So now, product of first 50 numbers will be xy=27 while product of first 100 numbers will be $\frac{y^2}{x}=27$. On solving we get y=9 and x=3.

So their sum = x+y=12.

Alternatively

Look at the product, it is 27 which is nothing but $3^3$and the series is all about products. So the numbers in the series will be full of threes. Look at the answer options for numbers which can be expressed as sum of 2 powers of 3. Only $12=3+3^2$is the possibility.

Series is $3,3^2,3^3,\frac{1}{3},\frac{1}{3^2},\frac{1}{3},3,3^2.............$