A geometric progression is one in which each term is equal to the previous term multiplied by some constant factor. If we know the nth term of the progression, g(n), and the factor r, we can write the other terms as
...
g(n+2) = g(n) * r^2
g(n+1) = g(n) * r^1
g(n) = g(n) * 1 = g(n) * r^0
g(n-1) = g(n) / r^1 = g(n) * r^-1
g(n-2) = g(n) / r^2 = g(n) * r^-2
...
Or, in general,
g(n+m) = g(n) * r^m
So, we have 3 terms of a geometric progression, and the middle term is unity, or 1. So, we have
g(n) = 1
g(n+1) = 1 * r^1 = r
g(n - 1) = 1 * r^-1 = 1/r
We don't know r, but we're told that these three terms add up to 3.5.
1 + r + 1/r = 3.5
Can we solve that for r?
r + 1/r = 2.5
r^2 + 1 = 2.5r
r^2 - 2.5r + 1 = 0
That's a quadratic equation; we can use the quadratic formula to solve for r.
r = {-(-2.5) ± √[(-2.5)^2 - 4*1*1]} / (2*1)
r = {2.5 ± √[6.25 - 4]} / 2
r = {2.5 ± √2.25} / 2
r = {2.5 ± 1.5} / 2
r = 1.25 ± 0.75
r = 2 OR r = 0.5
So, we've got two possibilities for the factor, r! What does that mean?
Well, let's see what happens if we use the first one, r = 2. We have
g(n) = 1
g(n+1) = 1 * 2 = 2
g(n-1) = 1 / 2 = 0.5
Now, what if we use the other option, r = 0.5?
g(n) = 1
g(n+1) = 1 * 0.5 = 0.5
g(n-1) = 1 / 0.5 = 2
So, we get the same values, just in reverse order! And since you're only interested in what the other two terms are, not the order they go in, either value of r is fine.
Comments
A geometric progression is one in which each term is equal to the previous term multiplied by some constant factor. If we know the nth term of the progression, g(n), and the factor r, we can write the other terms as
...
g(n+2) = g(n) * r^2
g(n+1) = g(n) * r^1
g(n) = g(n) * 1 = g(n) * r^0
g(n-1) = g(n) / r^1 = g(n) * r^-1
g(n-2) = g(n) / r^2 = g(n) * r^-2
...
Or, in general,
g(n+m) = g(n) * r^m
So, we have 3 terms of a geometric progression, and the middle term is unity, or 1. So, we have
g(n) = 1
g(n+1) = 1 * r^1 = r
g(n - 1) = 1 * r^-1 = 1/r
We don't know r, but we're told that these three terms add up to 3.5.
1 + r + 1/r = 3.5
Can we solve that for r?
r + 1/r = 2.5
r^2 + 1 = 2.5r
r^2 - 2.5r + 1 = 0
That's a quadratic equation; we can use the quadratic formula to solve for r.
r = {-(-2.5) ± √[(-2.5)^2 - 4*1*1]} / (2*1)
r = {2.5 ± √[6.25 - 4]} / 2
r = {2.5 ± √2.25} / 2
r = {2.5 ± 1.5} / 2
r = 1.25 ± 0.75
r = 2 OR r = 0.5
So, we've got two possibilities for the factor, r! What does that mean?
Well, let's see what happens if we use the first one, r = 2. We have
g(n) = 1
g(n+1) = 1 * 2 = 2
g(n-1) = 1 / 2 = 0.5
Now, what if we use the other option, r = 0.5?
g(n) = 1
g(n+1) = 1 * 0.5 = 0.5
g(n-1) = 1 / 0.5 = 2
So, we get the same values, just in reverse order! And since you're only interested in what the other two terms are, not the order they go in, either value of r is fine.
I hope that helps!
Let the other two terms be x and y. It then follows that:
x + y = 2.5 and 1/x = y
1/x + x = 2.5 => 1 + x² = 5x/2 => 2 + 2x² = 5x
2x² - 5x + 2 = 0
2x² - 4x - x + 2 = 0
(2x - 1)(x - 2) = 0
x = 1/2, 2
y = 2, 1/2
So the two terms can be x = 1/2, y = 2 or vice versa.
The three terms can be written as x-d, x, x+d
3x = 3.5
x = 3.5/3, the middle term.
So, it is impossible to have such an AP.