3 numbers A, B, C in that order, are in geometric progression with common ratio r. Given further that A, 2B, C in that order are in arithmetic progression, determine the possible values of r.

Question

3 numbers A, B, C in that order, are in geometric progression with common ratio r. Given further that A, 2B, C in that order are in arithmetic progression, determine the possible values of r.

Answer 1

a, ar, ar^2

a, a+d, a + 2d
well
2 a r = a + d
so
d = 2 a r - a = a (2r-1)
and
a + 2 d = a r^2
2 d =a(r^2-1)
so
2 d = a(r^2-1) = 2a(2r-1)
r^2-1=4r - 2
r^2- 4 r + 1 = 0

Answer 2

To solve this problem, let's use the properties of arithmetic and geometric progressions to find the values of r that satisfy the given conditions.

First, let's express the terms in terms of their predecessors in the sequence.

For the geometric progression, we can write:
B = Ar
C = Br^2

Similarly, for the arithmetic progression:
2B = A + C

Let's substitute the expressions for B, C, and 2B back into the equation 2B = A + C:

2(Ar) = A + (Ar^2)

Expand and simplify the equation:

2Ar = A + Ar^2

Divide both sides of the equation by A:

2r = 1 + r^2

Rearrange the equation to obtain a quadratic equation:

r^2 - 2r + 1 = 0

Now, we can solve this quadratic equation by factoring or by using the quadratic formula. Let's use factoring:

(r - 1)(r - 1) = 0

Since (r - 1)(r - 1) = 0, we have only one possible value for r:

r = 1

Therefore, the only possible value of r that satisfies the given conditions is 1.

Answer 3

To determine the possible values of r, we need to find the relationship between the terms in the geometric progression and the terms in the arithmetic progression.

Let's write the terms of the geometric progression as A, B, and C. We know that B/A = C/B = r.

Now let's write the terms of the arithmetic progression as A, 2B, and C. We know that 2B - A = C - 2B.

Rewriting this equation, we get:

2B - A = C - 2B
4B - 2A = 2C - 4B
6B - 3A = 3C - 6B

Now, let's substitute B/A = C/B = r into the equation:

6(rA) - 3A = 3(rC) - 6(rA)

Expanding and rearranging the equation, we get:

6rA - 3A = 3rC - 6rA
12rA = 3A + 3rC
12r = 3 + 3rC/A

Since A, B, and C are in geometric progression, we can write C/A = r^2. Substituting this into the equation, we get:

12r = 3 + 3r(r^2)
12r = 3 + 3r^3

Rearranging the equation, we get a cubic equation:

3r^3 - 12r + 3 = 0

To find the possible values of r, we can solve this cubic equation. However, solving this cubic equation is beyond the scope of a step-by-step bot. You can use numerical methods, such as Newton's method or a graphing calculator, to approximate the values of r.