Question

IF the G.P. of first six terms of any G.P is equal to 9 times the sum of first three terms then find the commen ratio of the G.P.

Answer 1

Solution 1

Consider a GP as
a,ar,ar2,ar3,ar4,ar5a,ar,ar2,ar3,ar4,ar5

Sum of first 3 terms 
=a+ar+ar2⋯(1)=a+ar+ar2⋯(1) 

Sum of first 6 terms 
=a+ar+ar2+ar3+ar4+ar5=a+ar+ar2+r3(a+ar+ar2)⋯(2)=a+ar+ar2+ar3+ar4+ar5=a+ar+ar2+r3(a+ar+ar2)⋯(2)

Let a+ar+ar2=xa+ar+ar2=x.
Then,
Sum of first 3 terms =x=x
Sum of first 6 terms =x+r3x=x(r3+1)=x+r3x=x(r3+1)

x(r3+1):x=9:1(r3+1):1=9:1r3+1=9r3=8r=2