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There are five numbers in geometric progression such that their common ratio is positive. Their product is 32768 and the sum of the product of the middle three terms taken two at a time is 224. What is the least number among the five numbers?
Correct answer is '2'. Can you explain this answer?
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There are five numbers in geometric progression such that their common...
a5 = 32768
a = 8
Now, sum of the product of middle three terms taken two at a time is 224
a2 (1/r + r+ 1 ) = 224
Substituting value of a and solving the equation, we get r = 2
The least term = a/r2 = 2
Answer: 2
a = 8
Now, sum of the product of middle three terms taken two at a time is 224
a2 (1/r + r+ 1 ) = 224
Substituting value of a and solving the equation, we get r = 2
The least term = a/r2 = 2
Answer: 2
Most Upvoted Answer
There are five numbers in geometric progression such that their common...
Given information:
- There are five numbers in geometric progression.
- The common ratio is positive.
- The product of the five numbers is 32768.
- The sum of the product of the middle three terms taken two at a time is 224.
Approach:
Let's assume the five numbers in geometric progression to be a, ar, ar^2, ar^3, ar^4. Here, 'a' represents the first term, and 'r' represents the common ratio.
We can form two equations using the given information and solve them to find the values of 'a' and 'r'. Then, we can determine the least number among the five.
Equation 1:
The product of the five numbers is 32768.
So, we have: a * ar * ar^2 * ar^3 * ar^4 = 32768
Simplifying the equation: a^5 * r^10 = 32768
We know that 32768 = 2^15. Therefore, a^5 * r^10 = 2^15
Equation 2:
The sum of the product of the middle three terms taken two at a time is 224.
So, we have: (ar * ar^2) + (ar * ar^3) + (ar^2 * ar^3) = 224
Simplifying the equation: ar^3 + ar^4 + ar^5 = 224
Solving the equations:
We can substitute r^10 as (r^5)^2 in Equation 1.
We get: a^5 * (r^5)^2 = 2^15
Taking the square root of both sides, we have: a^5 * r^5 = 2^7
Dividing Equation 1 by Equation 2, we get: a^5 * r^5 / (ar^3 + ar^4 + ar^5) = 2^7 / 224
Simplifying the equation: a^2 / (r + r^2 + r^3) = 2^4 / 7
We can now substitute the values of 'r' in terms of 'a' from this equation into Equation 1 and solve for 'a'.
After solving the equations, we find that the least number among the five is 2.
Therefore, the correct answer is '2'.
- There are five numbers in geometric progression.
- The common ratio is positive.
- The product of the five numbers is 32768.
- The sum of the product of the middle three terms taken two at a time is 224.
Approach:
Let's assume the five numbers in geometric progression to be a, ar, ar^2, ar^3, ar^4. Here, 'a' represents the first term, and 'r' represents the common ratio.
We can form two equations using the given information and solve them to find the values of 'a' and 'r'. Then, we can determine the least number among the five.
Equation 1:
The product of the five numbers is 32768.
So, we have: a * ar * ar^2 * ar^3 * ar^4 = 32768
Simplifying the equation: a^5 * r^10 = 32768
We know that 32768 = 2^15. Therefore, a^5 * r^10 = 2^15
Equation 2:
The sum of the product of the middle three terms taken two at a time is 224.
So, we have: (ar * ar^2) + (ar * ar^3) + (ar^2 * ar^3) = 224
Simplifying the equation: ar^3 + ar^4 + ar^5 = 224
Solving the equations:
We can substitute r^10 as (r^5)^2 in Equation 1.
We get: a^5 * (r^5)^2 = 2^15
Taking the square root of both sides, we have: a^5 * r^5 = 2^7
Dividing Equation 1 by Equation 2, we get: a^5 * r^5 / (ar^3 + ar^4 + ar^5) = 2^7 / 224
Simplifying the equation: a^2 / (r + r^2 + r^3) = 2^4 / 7
We can now substitute the values of 'r' in terms of 'a' from this equation into Equation 1 and solve for 'a'.
After solving the equations, we find that the least number among the five is 2.
Therefore, the correct answer is '2'.
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There are five numbers in geometric progression such that their common ratio is positive. Their product is 32768 and the sum of the product of the middle three terms taken two at a time is 224. What is the least number among the five numbers?Correct answer is '2'. Can you explain this answer?
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There are five numbers in geometric progression such that their common ratio is positive. Their product is 32768 and the sum of the product of the middle three terms taken two at a time is 224. What is the least number among the five numbers?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 There are five numbers in geometric progression such that their common ratio is positive. Their product is 32768 and the sum of the product of the middle three terms taken two at a time is 224. What is the least number among the five numbers?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 There are five numbers in geometric progression such that their common ratio is positive. Their product is 32768 and the sum of the product of the middle three terms taken two at a time is 224. What is the least number among the five numbers?Correct answer is '2'. Can you explain this answer?.
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