5. In an A. P, the sum of three consecutive terms is 24. When 1 is subtracted to the first term, 2 to the

second, then the terms form a G. P. Find the terms of the A. P.

Step-by-step explanation:

Let the three numbers be a-d, a, a+d being a series in AP.

The sum of the three numbers is 24 = a-d + a + a+d = 3a

Thus a = 8.

Now a-d-1, a-2, a+d form a GP. It means, the common ratios should be equal or the middle term should be the Geometric Mean of the first and third terms, or

(a-d-1) (a+d) = (a-2) ^2 or

(8-d-1) (8+d) = (8-2) ^2, or

(7-d) (8+d) = 36, or

56 - 8d + 7d - d^2 = 36, or

56 - 8d + 7d - d^2 - 36 = 0, or

d^2 + d - 20 = 0

(d+5) (d-4) = 0

Thus d = - 5 or 4

So the AP is 13,8,3 or 4,8,12

Check: 13,8,3 in AP becomes (13-1), (8-2), 3 = 12, 6, 3 which is a GP with r = (1/2)

4,8,12 in AP becomes (4-1), (8-2), 12 = 3, 6, 12 which is a GP with r = 2