Three numbers form a geometric progression. if the second term is increased by 2, then the progression will become arithmetic and if, after this, the last term is increased by 9, then the progression will again become geometric. find these three numbers.

The three terms in geometric progression are 4, 8, and 16 and this can be determined by using the properties of geometric progression.

Given :

Let the three terms in geometric progression be a, ar, and . Given that if the second term is increased by 2 then the progression will become arithmetic that is:

and

Now, equate both the equation.

Factorize the above equation.

   ---- (1)

Now, it is also given that the last term is increased by 9, that is:

a, ar + 2, a + 9

The common ratio of the above geometric progression is:

  ---- (2)

 ---- (3)

Now, equate both the equation.

Cross multiply in the above equation.

9a - 4ar - 4 = 0

  --- (4)

Now, equate equation (1) and equation (4).

Cross multiply in the above equation.

Factorize the above equation.

(r + 4)(r - 2) = 0

r = 2 (Accepted)

Now, put the value of r in the equation (4).

a = 4

ar = 4(2) = 8

a = 4(4) = 16

The three terms in geometric progression are 4, 8, and 16.

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