Question

If the first term minus third term of a G. P. is 768 and the third term minus seventh term of the same G. P. is 240, then the product of first 21 terms is 

Answer 1

let first term=a and common ratio=r.
therefore a-ar^2 = 768 => a(1-r^2) = 768.
also ar^2-ar^6 = 240 => ar^2(1-r^4) = 240.
dividing the 2 equations we get r^2(1+r^2) = 0.3125 => r^2 = 0.25 (only legitimate solution).
hence r = +1/2 or -1/2. hence a = 1024 = 2^10.
therefore product of first 21 terms = (a^2*r^20)^10*a*r^10 = a^21*r^210 = 2^210*(1/2)^210 = 1.

Answer 2

 a-ar^2 = 768 => a(1-r^2) = 768.
 ar^2-ar^6 = 240 => ar^2(1-r^4) = 240.
 r^2(1+r^2) = 0.3125 => r^2 = 0.25 (only legitimate solution).
 r = +1/2 or -1/2. hence a = 1024 = 2^10.
 = (a^2*r^20)^10*a*r^10 = a^21*r^210 = 2^210*(1/2)^210 = 1