The numbers p,10 and q are 3 consecutive terms of an arithmetic progression .the numbers p,6 and q are 3 consecutive terms of a geometric progression .by first forming two equations in p and q show that p^2-20p+36=0
Hence find the values of p and q for which the geometric progression converges
Help please
from the first:
10-p = q-10
q = 20-p
from the second:
pq = 36
by substitution,
p(20-p) = 36
20p - p^2 = 36
p^2 - 20p + 36 = 0 ----> thus shown
(p - 18)(p - 2) = 0
p = 18 or p = 2
then q = 36/18 = 2 or p = 36/2 = 18
for p = 18, q = 2
the AS is 18, 10 and 2
the GS is 18, 6, 2 -----> which converges, r = 1/2
for p = 2, q = 18
the AS is 2, 10, 18
the GS is 2, 6, 18, ---> diverges , r > 1
i want specifically for P,6andQ
To solve this problem, let's first consider the arithmetic progression. We know that the terms p, 10, and q form an arithmetic progression, which means the difference between consecutive terms is constant. Let's define this common difference as d.
Based on this information, we can write the following equation:
q = p + 2d
Now, let's consider the geometric progression. We know that the terms p, 6, and q form a geometric progression, which means the ratio between consecutive terms is constant. Let's define this common ratio as r.
Based on this information, we can write the following equation:
q = p * r
Now, we have two equations involving p and q. Let's substitute the value of q from the first equation into the second equation:
p + 2d = p * r
Rearranging this equation, we get:
p * r - p = 2d
Factoring out p on the left-hand side, we get:
p(r - 1) = 2d
Dividing both sides by (r - 1), we get:
p = 2d / (r - 1)
Now, substitute the value of p into the equation q = p + 2d:
q = (2d / (r - 1)) + 2d
q = (2d + 2d(r - 1)) / (r - 1)
q = (2d + 2dr - 2d) / (r - 1)
q = 2dr / (r - 1)
Now, let's rearrange this equation to get r in terms of p:
q = 2dr / (r - 1)
Multiply both sides by (r - 1):
q(r - 1) = 2dr
Expand the equation:
qr - q = 2dr
Rearrange the equation:
qr - 2dr = q
Factor out r on the left-hand side:
r(q - 2d) = q
Divide both sides by (q - 2d):
r = q / (q - 2d)
Now, we have two expressions for r. We can equate them:
2d / (r - 1) = q / (q - 2d)
Cross-multiply:
2d(q - 2d) = q(r - 1)
Expand both sides:
2dq - 4d^2 = qr - q
Rearrange the equation:
qr - 2dq = q - 4d^2
Now, notice that we can substitute q = pr into the equation:
pr - 2dp = p - 4d^2
Rearrange the equation:
pr - p = 2dp - 4d^2
Now, let's simplify the equation and solve for p:
p(r - 1) = d(2p - 4d)
Rearrange the equation:
p(r - 1) = 2pd - 4d^2
Divide both sides by (r - 1):
p = (2pd - 4d^2) / (r - 1)
Now, substitute the value of p into the equation p = 2d / (r - 1):
(2d / (r - 1))(r - 1) = (2d - 4d^2) / (r - 1)
Cancel out the (r - 1) terms:
2d = 2d - 4d^2
Rearrange the equation:
4d^2 = 0
Divide both sides by 4:
d^2 = 0
Take the square root of both sides:
d = 0
Now, substitute the value of d = 0 into the equation q = p + 2d:
q = p + 2(0)
q = p
Substitute the value of q = p into the equation q = p * r:
p = p * r
Divide both sides by p:
1 = r
Therefore, for the geometric progression to converge, r must equal 1. Substituting this value of r = 1 into the equation q = p * r:
q = p * 1
q = p
So, p and q can take any value for which the arithmetic progression has a common difference of 0 (i.e., p = q).
To find the values of p and q, we need to solve two equations: one from the arithmetic progression and one from the geometric progression.
1. Equation from the arithmetic progression:
In an arithmetic progression, the difference between consecutive terms is constant.
So, we can write the following equation using p, 10, and q:
10 - p = q - 10
2. Equation from the geometric progression:
In a geometric progression, the ratio of two consecutive terms is constant.
So, we can write the following equation using p, 6, and q:
q/6 = 6/p
Now, we have two equations. We can solve them simultaneously to find the values of p and q.
From equation 2, we can rewrite it as q = (6^2)/p.
Substituting this value of q in equation 1, we get:
10 - p = (6^2)/p - 10
Multiplying both sides of the equation by p, we get:
10p - p^2 = 36 - 10p
Rearranging the terms, we get:
p^2 - 20p + 36 = 0
This is a quadratic equation in p. Now, we can solve this equation to find the values of p.
To solve the quadratic equation, we can use the quadratic formula:
p = (-b ± √(b^2 - 4ac)) / (2a)
Comparing the equation p^2 - 20p + 36 = 0 with the standard form ax^2 + bx + c = 0, we have:
a = 1, b = -20, c = 36.
Substituting these values in the quadratic formula, we get:
p = (-(-20) ± √((-20)^2 - 4(1)(36))) / (2(1))
Simplifying this equation, we get:
p = (20 ± √(400 - 144)) / 2
p = (20 ± √256) / 2
p = (20 ± 16) / 2
There are two possible values for p:
1) p = (20 + 16) / 2 = 36 / 2 = 18
2) p = (20 - 16) / 2 = 4 / 2 = 2
Now that we have the values of p, we can substitute them back into the equations to find the corresponding values of q.
For p = 18:
From equation 2, we have q = (6^2)/18 = 2
For p = 2:
From equation 2, we have q = (6^2)/2 = 18
Therefore, the values of p and q for which the geometric progression converges are p = 18 and q = 2, as well as p = 2 and q = 18.