Question

Three positive number whose sum is 21 are in arithmetic progression, If 2, 2,14 are added to them added to them respectively then resulting number are in geometric progression .Then which of the following is not among the three number ?
1. 25
2. 13
3. 1
4. 7

Answer 1

Correct Answer - Option 1 : 25

Concept:

If a, b, c are in GP then \(\rm \dfrac b a = \dfrac c b\)

Calculations:

Consider, three positive number  a - d, a, a+ d are in AP

Three positive number whose sum is 21.

⇒a - d + a + a+ d = 21 

⇒ a = 7

Hence, three positive number becomes 7 - d, 7, 7+ d.

It is given, If 2, 2,14 are added to them respectively then resulting number are in geometric progression.

⇒ 9 - d, 9, 21+ d are in GP.

⇒ \(\rm \dfrac {9}{9-d}=\dfrac {21+d}{9}\)

⇒ 81 = (21 + d) (9 - d)

⇒ \(\rm d^2 +12d - 108 = 0\)

⇒ (d + 18)(d - 6) = 0 

⇒ d = - 18 and d = 6

When d = - 18, the three numbers are 25, 7, -11.

When d = 6, the three numbers are 1, 7, 13.