Correct Answer - Option 4 : 3/2
Concept:
If a, ar, ar2, ar3,.....,arn-1 are in GP then nth term of GP is given by Tn = arn-1
If a1, a2, a3, a4 are in GP, then common ratio of GP is given by, \( r=\frac{a_{2}}{a_{1}}=\frac{a_{3}}{a_{2}}=\frac{a_{4}}{a_{3}}\)
Calculation:
Lets three terms of GP be a/r, a, ar
Given: The product of first three terms of GP = 216 and their sum is 19
\(⇒ \rm \frac{a}{r}×{a}×{ar}=125\) ⇒ a3 = 216 ⇒ a = 6
According to second statement,
\(⇒ \rm \frac{a}{r}+{a}+{ar}=19⇒ a\left (\frac{1}{r}+1+r \right )=19\)
Put the value of a = 6 in the above equation
\(⇒ \rm 6\left (\frac{1}{r}+1+r \right )=19\)
⇒ 6 + 6r + 6r2 = 19r ⇒ 6r2 - 13r + 6 = 0
⇒ (3r - 2) × (2r - 3) = 0
⇒ r = 2/3 or 3/2
As it is given that r > 1
Hence common ratio r = 3/2.