UPSC NDA Math Model Paper 5

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.
⇒

9

9−d

=

21+d

9

⇒81=(21+d)(9−d)
⇒d2+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.