In a GP, the ratio of the sum of the first three terms is to first six terms is 125 : 152. Find the common ratio.
Sum of a G.P. series is represented by the formula, , when r≠1. ‘Sn’ represents the sum of the G.P. series upto nth terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.
Sum of first 3 terms =
Sum of first 6 terms =
∴
⇒
⇒ 152r3 – 152= 125r6-125
⇒125r6-152r3-125+152 = 0
⇒125r6 – 152r3 + 27 = 0
⇒125r6 – 125r3 – 27r3 + 27 = 0
⇒ (125r3 - 27) (r3-1)= 0
Either 125r3 -27 = 0 or r3-1 = 0
Either 125r3=27 or r3=1
Either r3 = or r=1
Either r= or r=1
Since r ≠ 1 [ if r is 1, all the terms will be equal which destroys the purpose ]
∴ r =