Question

If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.

Answer 1

Given: Second, third and sixth terms of an A.P. are consecutive terms of a G.P. 

Let the first term of AP be a and the common difference be d. 

⇒ A_n = a+(n-1)d 

⇒ A₂ = a+d 

⇒ A₃ = a+2d 

⇒ A₆ = a+5d 

If a,b,c are consecutive terms of GP then we can write b² = a.c 

∴ We can write (a+2d)² = (a+d).(a+5d) 

⇒ a²+4d²+4ad = a²+6ad+5d² 

⇒ d²+2ad = 0 

⇒ d(d+2a) =0 

∴ d = 0 or d =-2a 

When d = 0 then the GP becomes a,a,a. 

∴ The common ration becomes 1. 

When d = -2a then the GP becomes –a, -3a,-9a 

∴ The common ratio becomes 3.