Question

The ratio of the sum of first three terms is to that of first six terms of a GP is 125 : 152. Find the common ratio.

Answer 1

The first three terms of a G.P. are: a,ar,ar² 

The first six terms of a G.P. are: a,ar,ar² ,ar³ ,ar⁴ ,ar⁵ 

It is given that the ratio of the sum of first three terms is to that of first six terms of a GP is 125 : 152. 

⇒ a + ar + ar² = 125x & a + ar + ar² + ar³ + ar⁴ + ar⁵ = 152x 

⇒ a + ar + ar² + r³ ( a + ar + ar² ) = 152x 

⇒ 125x + r³ (125x) = 152x 

⇒ r³ (125x) = 152x - 125x = 27x 

⇒ r³ = \(\frac{27}{125} = \left(\frac{3}{5}\right)^3\)

⇒ r = 3/5

Hence common ratio is 3/5