Q. 93.5( 13 Votes )

The sum of three numbers in G.P. is 21, and the sum of their squares is 189. Find the numbers.

Answer :

Let the three numbers be a, ar, and ar2


According to the question


a + ar + ar2 = 21


a(1 + r + r2) = 21


Squaring both sides we get,


a2(1 + r + r2)2 = (21)2….(1)


And from the second condition,


a2 + a2r2 + a2r4 = 189


a2(1 + r2 + r4) = 189……(2)


Dividing both the equations we get,




Cross multiplying we get,


3 + 3r + 3r2 = 7r2 – 7r + 7


4r2 – 10r + 4 = 0


2r2 – 5r + 2 = 0


Factorizing the quadratic equation such that, on multiplication, we get 4 and on the addition, we get 5. So,


2r2 – (4r + r) + 2 = 0


2r(r – 2) –1(r – 2) = 0


(2r – 1)(r – 2) = 0


r = 1/2 , r = 2


Putting the value of r in equation 2 we get,


At r = 2,


a2(1 + r2 + r4) = 189


a2(1 + 4 + 16) = 189


a2


a2 = 9


a = ±3


At r = 1/2





a2 = 9 × 16


a = 3 × 4 = 12


The numbers are:


1, 9, 81 or 81, 9, 1


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