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Let x, y, z be three positive real numbers in a geometric progression such that x < y < z. If 5x, 16y, and 12z are in an arithmetic progression then the common ratio of the geometric progression is
- a)3/6
- b)1/6
- c)5/2
- d)3/2
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Let x, y, z be three positive real numbers in a geometric progression ...
Let x = a, y = ar and z = ar2
It is given that, 5x, 16y and 12z are in AP.
so, 5x + 12z = 32y
On replacing the values of x, y and z, we get
5a + 12ar2 = 32ar
or, 12r2 – 32r + 5 = 0
On solving, r =5/2 or 1/6
For r = 1/6, x < y < z is not satisfied.
So, r=5/2
Hence, option C is the correct answer.
Most Upvoted Answer
Let x, y, z be three positive real numbers in a geometric progression ...
Let x = a, y = ar and z = ar2
It is given that, 5x, 16y and 12z are in AP.
so, 5x + 12z = 32y
On replacing the values of x, y and z, we get
5a + 12ar2 = 32ar
or, 12r2 – 32r + 5 = 0
On solving, r =5/2 or 1/6
For r = 1/6, x < y < z is not satisfied.
So, r=5/2
Hence, option C is the correct answer.
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Community Answer
Let x, y, z be three positive real numbers in a geometric progression ...
Understanding the Problem
Given three positive real numbers x, y, z in geometric progression (GP), we know that:
- y = xr (where r is the common ratio)
- z = xr^2
With the condition that 5x, 16y, and 12z are in arithmetic progression (AP), we can express this condition mathematically as:
Arithmetic Progression Condition
The AP condition implies:
16y - 5x = 12z - 16y
Substituting the expressions for y and z gives us:
16(xr) - 5x = 12(xr^2) - 16(xr)
This simplifies to:
16xr - 5x = 12xr^2 - 16xr
Rearranging the Equation
Combining like terms results in:
(16xr + 16xr - 5x) = 12xr^2
This can be simplified to:
32xr - 5x = 12xr^2
Factoring Out x
Factoring x out from the left side leads to:
x(32r - 5) = 12xr^2
Dividing both sides by x (since x is positive) results in:
32r - 5 = 12r^2
Rearranging to Form a Quadratic Equation
Rearranging gives:
12r^2 - 32r + 5 = 0
Using the Quadratic Formula
Applying the quadratic formula:
r = [32 ± sqrt((32)^2 - 4 * 12 * 5)] / (2 * 12)
Calculating the discriminant:
= 1024 - 240 = 784
The roots are:
r = (32 ± 28) / 24
This gives two possible values for r:
1. r = 60 / 24 = 2.5
2. r = 4 / 24 = 1/6
Since we require the common ratio to be positive and consistent with x < y="" />< z,="" we="" />
Final Answer
The common ratio of the geometric progression is:
r = 5/2 (Option C)
Given three positive real numbers x, y, z in geometric progression (GP), we know that:
- y = xr (where r is the common ratio)
- z = xr^2
With the condition that 5x, 16y, and 12z are in arithmetic progression (AP), we can express this condition mathematically as:
Arithmetic Progression Condition
The AP condition implies:
16y - 5x = 12z - 16y
Substituting the expressions for y and z gives us:
16(xr) - 5x = 12(xr^2) - 16(xr)
This simplifies to:
16xr - 5x = 12xr^2 - 16xr
Rearranging the Equation
Combining like terms results in:
(16xr + 16xr - 5x) = 12xr^2
This can be simplified to:
32xr - 5x = 12xr^2
Factoring Out x
Factoring x out from the left side leads to:
x(32r - 5) = 12xr^2
Dividing both sides by x (since x is positive) results in:
32r - 5 = 12r^2
Rearranging to Form a Quadratic Equation
Rearranging gives:
12r^2 - 32r + 5 = 0
Using the Quadratic Formula
Applying the quadratic formula:
r = [32 ± sqrt((32)^2 - 4 * 12 * 5)] / (2 * 12)
Calculating the discriminant:
= 1024 - 240 = 784
The roots are:
r = (32 ± 28) / 24
This gives two possible values for r:
1. r = 60 / 24 = 2.5
2. r = 4 / 24 = 1/6
Since we require the common ratio to be positive and consistent with x < y="" />< z,="" we="" />
Final Answer
The common ratio of the geometric progression is:
r = 5/2 (Option C)
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