Question

Three integers form an increasing GP.If 3rd nos decreased by 16,we get an AP.If then the 2nd nos decreased by 2,we again get a GP.Find the 3 nos

Answer 1

Let the first three terms of GP be a,ar,$ar^2$.

If third number is decreased by 16,we get an AP.

Therefore, the terms are a, ar, $ar^2 - 16$

Since these are in AP, therefore the common difference will be same.

So, $ar^2-16-ar = ar-a$

$ar^2-2ar+a=16$

Dividing by 'a', we get

$r^2-2r-15 = 0$

$r^2-5r+3r-15 = 0$

$r(r-5)+3(r-5)=0$

(r-5)(r+3)=0

So, r= 5 and -3

Therefore, r = 5

Now, $ar^2-2ar+a=16$

Putting the value of r = 5, we get

$25a-10a+a=16$

So, a =1

Therefore, the numbers are

1, 5, 25.

Hence, the required numbers are 1,5 and 25.