The first term of an AP is 12 and that of a GP is 6. The common difference and the common ratio are equal, so as are the sum of the fist 3 terms. Find the possible values of the common difference?
12 + 12+d + 12+2d = 6(1+r+r^2)
But, d=r, so
36+3d = 6(1+d+d^2)
6d^2+3d-30 = 0
2d^2 + d - 10 = 0
(2d+5)(d-2) = 0
So, d=2 or -5/2
The two sequences are thus
12,14,16
6,12,24
with S3=42
or,
12,9.5,7
6,-15,75/2
with S3=28.5
To find the possible values of the common difference in the arithmetic progression (AP), we need to set up the equation with the given information.
Let's assume that the common difference in the AP is "d".
The first term of the AP is given as 12, so the second term will be 12 + d, the third term will be 12 + 2d, and so on.
Now, let's look at the given information about the geometric progression (GP). The first term of the GP is 6, and the common ratio is also "d".
Since the sum of the first three terms of both the AP and the GP is equal, we can set up the following equations:
AP: 12 + (12 + d) + (12 + 2d) = 6 + 6d + 6d^2
We can simplify this equation as follows:
36 + 3d = 6 + 12d + 6d^2
Rearranging the terms, we have:
6d^2 + 9d - 30 = 0
Dividing by 3 to simplify the equation, we get:
2d^2 + 3d - 10 = 0
Now we can factor or use the quadratic formula to solve for "d".
The factorized form of the equation is:
(2d - 5)(d + 2) = 0
From this equation, we can see that there are two possible values for "d":
d = 5/2 or d = -2
Therefore, the possible values of the common difference in the arithmetic progression are 5/2 or -2.