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Three distinct numbers a, b and c are in geometric progression, and their cyclic sums, taking two at a time, a + b, b + c and c + a are in arithmetic progression. What is the common ratio of the geometric progression involving a, b and c?
Correct answer is '-2'. Can you explain this answer?
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Three distinct numbers a, b and c are in geometric progression, and th...
Common ratio of the geometric progression = c/b = -2 Answer: -2
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Three distinct numbers a, b and c are in geometric progression, and th...
Given:
Three distinct numbers a, b, and c are in geometric progression.
The cyclic sums, taking two at a time, a b, b c, and c a are in arithmetic progression.
To find:
The common ratio of the geometric progression involving a, b, and c.
Solution:
Let the three numbers be a, ar, and ar^2, where r is the common ratio of the geometric progression.
The cyclic sums are given as follows:
a b = a + ar
b c = ar + ar^2
c a = ar^2 + a
We are given that the cyclic sums are in arithmetic progression. So, let's assume the common difference of the arithmetic progression is d.
Therefore, we have the following equations:
b - a = d
c - b = d
Simplifying the equations, we get:
ar - a = d
ar^2 - ar = d
Step 1: Solve the first equation for a:
ar - a = d
a(r - 1) = d
a = d / (r - 1) ----(1)
Step 2: Solve the second equation for a:
ar^2 - ar = d
ar(r - 1) = d
r(r - 1) = d / a
Substituting the value of a from equation (1) into the above equation, we get:
r(r - 1) = d / (d / (r - 1))
r(r - 1) = (r - 1)
r^2 - r - 1 = 0
Step 3: Solve the quadratic equation:
Using the quadratic formula, we have:
r = (-(-1) ± √((-1)^2 - 4(1)(-1))) / (2(1))
r = (1 ± √(1 + 4)) / 2
r = (1 ± √5) / 2
Since the numbers are distinct, we can eliminate the negative value. Therefore, the possible value of r is:
r = (1 + √5) / 2
Step 4: Verify the value of r:
Let's substitute the value of r into the cyclic sums and check if they are in arithmetic progression:
a + ar = a(1 + r)
= a(1 + (1 + √5) / 2)
= a(2 + √5) / 2
ar + ar^2 = ar(1 + r)
= ar(1 + (1 + √5) / 2)
= ar(2 + √5) / 2
ar^2 + a = ar^2 + a(1)
= ar^2 + a(1 + √5) / 2
= ar^2 + a(2 + √5) / 2
We can observe that all three expressions have a common factor of (2 + √5) / 2, which simplifies to 1 + √5. Therefore, the cyclic sums are in arithmetic progression.
Conclusion:
The common ratio of the geometric progression involving a, b, and c is (1 +
Three distinct numbers a, b, and c are in geometric progression.
The cyclic sums, taking two at a time, a b, b c, and c a are in arithmetic progression.
To find:
The common ratio of the geometric progression involving a, b, and c.
Solution:
Let the three numbers be a, ar, and ar^2, where r is the common ratio of the geometric progression.
The cyclic sums are given as follows:
a b = a + ar
b c = ar + ar^2
c a = ar^2 + a
We are given that the cyclic sums are in arithmetic progression. So, let's assume the common difference of the arithmetic progression is d.
Therefore, we have the following equations:
b - a = d
c - b = d
Simplifying the equations, we get:
ar - a = d
ar^2 - ar = d
Step 1: Solve the first equation for a:
ar - a = d
a(r - 1) = d
a = d / (r - 1) ----(1)
Step 2: Solve the second equation for a:
ar^2 - ar = d
ar(r - 1) = d
r(r - 1) = d / a
Substituting the value of a from equation (1) into the above equation, we get:
r(r - 1) = d / (d / (r - 1))
r(r - 1) = (r - 1)
r^2 - r - 1 = 0
Step 3: Solve the quadratic equation:
Using the quadratic formula, we have:
r = (-(-1) ± √((-1)^2 - 4(1)(-1))) / (2(1))
r = (1 ± √(1 + 4)) / 2
r = (1 ± √5) / 2
Since the numbers are distinct, we can eliminate the negative value. Therefore, the possible value of r is:
r = (1 + √5) / 2
Step 4: Verify the value of r:
Let's substitute the value of r into the cyclic sums and check if they are in arithmetic progression:
a + ar = a(1 + r)
= a(1 + (1 + √5) / 2)
= a(2 + √5) / 2
ar + ar^2 = ar(1 + r)
= ar(1 + (1 + √5) / 2)
= ar(2 + √5) / 2
ar^2 + a = ar^2 + a(1)
= ar^2 + a(1 + √5) / 2
= ar^2 + a(2 + √5) / 2
We can observe that all three expressions have a common factor of (2 + √5) / 2, which simplifies to 1 + √5. Therefore, the cyclic sums are in arithmetic progression.
Conclusion:
The common ratio of the geometric progression involving a, b, and c is (1 +
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Three distinct numbers a, b and c are in geometric progression, and their cyclic sums, taking two at a time, a + b, b + c and c + a are in arithmetic progression. What is the common ratio of the geometric progression involving a, b and c?Correct answer is '-2'. Can you explain this answer?
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Three distinct numbers a, b and c are in geometric progression, and their cyclic sums, taking two at a time, a + b, b + c and c + a are in arithmetic progression. What is the common ratio of the geometric progression involving a, b and c?Correct answer is '-2'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Three distinct numbers a, b and c are in geometric progression, and their cyclic sums, taking two at a time, a + b, b + c and c + a are in arithmetic progression. What is the common ratio of the geometric progression involving a, b and c?Correct answer is '-2'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Three distinct numbers a, b and c are in geometric progression, and their cyclic sums, taking two at a time, a + b, b + c and c + a are in arithmetic progression. What is the common ratio of the geometric progression involving a, b and c?Correct answer is '-2'. Can you explain this answer?.
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Three distinct numbers a, b and c are in geometric progression, and their cyclic sums, taking two at a time, a + b, b + c and c + a are in arithmetic progression. What is the common ratio of the geometric progression involving a, b and c?Correct answer is '-2'. Can you explain this answer?, a detailed solution for Three distinct numbers a, b and c are in geometric progression, and their cyclic sums, taking two at a time, a + b, b + c and c + a are in arithmetic progression. What is the common ratio of the geometric progression involving a, b and c?Correct answer is '-2'. Can you explain this answer? has been provided alongside types of Three distinct numbers a, b and c are in geometric progression, and their cyclic sums, taking two at a time, a + b, b + c and c + a are in arithmetic progression. What is the common ratio of the geometric progression involving a, b and c?Correct answer is '-2'. Can you explain this answer? theory, EduRev gives you an
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