# If the product of first three terms of a GP is 216 and their sum is 19, then find common ratio r such that r > 1 .

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If the product of first three terms of a GP is 216 and their sum is 19, then find common ratio r such that r > 1 .
1. 7/6
2. 13/6
3. 2/3
4. 3/2
5. None of these

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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.