Question

For a G.P., if the sum of the first 3 terms is 125 and the sum of the next 3 terms is 27, find the value of r.

Answer 1

S₃ = 125, S₆ = 125 + 27 = 152

S_n = `"a"((1 - "r"^"n")/(1 - "r"))`

∴ S₃ = `"a"((1 - "r"^3)/(1 - "r"))`

∴ 125 = `"a"((1 - "r"^3)/(1 - "r"))`       ...(i)

Also, S₆ = `"a"((1 - "r"^6)/(1 - "r"))`

∴ 152 = `"a"((1 - "r"^6)/(1 - "r"))`       ...(ii)
Dividing (ii) by (i), we get

`152/125 = (1 - "r"^6)/(1 - "r"^3)`

∴ `152/125 = ((1 + "r"^3)(1 - "r"^3))/((1 - "r"^3)`

∴ 1 + r³ = `152/125`

∴ r³ = `152/125 - 1`

∴ r³ = `27/125`

∴ r³ = `(3/5)^3`

∴ r = `3/5`