Three numbers are in A.P., and their sum is 15. If 1, 3, 9 be added to them respectively, they from a G.P. find the numbers.

Solution:

Consider the first term of an A.P. to be ‘a’ and let its common difference be‘d’.

We have, a₁ + a₂ + a₃ = 15

Where the three numbers are as follows:

a, a + d, and a + 2d

So, we can write:

Now, according to the question:

a + 1, a + d + 3, and a + 2d + 9

they are in GP, that is:

Then,

For a = 3 and d = 2, the A.P is 3, 5, 7

For a = 15 and d = -10, the A.P is 15, 5, -5

∴ The numbers are 3, 5, 7 or 15, 5, – 5