The first three terms of an arithmetic-geometric progression are 3, –1 and –1. The next term of the progression is
To find the next term of the arithmetic-geometric progression, we need to identify the common difference (d) for the arithmetic part and the common ratio (r) for the geometric part.
In an arithmetic-geometric progression, the arithmetic part represents a sequence where the difference between consecutive terms is constant (arithmetic), and the geometric part represents a sequence where the ratio between consecutive terms is constant (geometric).
Given that the first three terms are 3, -1, and -1, we can determine the common difference and common ratio.
To find the common difference:
d = (-1) - 3 = -4
To find the common ratio:
We divide each term by the previous term:
(-1) / 3 = -1/3
Now that we have the common difference (d = -4) and the common ratio (r = -1/3), we can calculate the next term of the progression using the formulas:
For arithmetic sequences: an = a1 + (n - 1) * d
For geometric sequences: an = a1 * r^(n - 1)
Since we have an arithmetic-geometric sequence, we can use either formula. Let's use the geometric formula:
a4 = (-1) * (-1/3)^(4 - 1)
a4 = (-1) * (-1/3)^3
a4 = (-1) * (-1/27)
a4 = 1/27
Therefore, the next term of the arithmetic-geometric progression is 1/27.
ab = 3
(a+d)(br) = -1
(a+2d)(br^2) = -1
One solution is d = -2, a = -d/2, b = -6/d, r = 1/3
AP: 1, -1, -3
GP: 3, 1, 1/3