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The product of n positive numbers is unity, then their sum is :
- a)a positive integer
- b)divisible by n
- c)equal to n + 1/n
- d)never less than n
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The product of n positive numbers is unity, then their sum is :a)a pos...
Since, product of n positive number is unity.
⇒ x1 x2 x3 ....... xn = 1 ..(i)
Using A.M. ≥ GM
⇒ x1 x2 x3 ....... xn = 1 ..(i)
Using A.M. ≥ GM
⇒ 
Hence
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The product of n positive numbers is unity, then their sum is :a)a pos...
To prove that the sum of n positive numbers whose product is unity is never less than n, we can use the concept of the arithmetic mean and geometric mean inequality.
Arithmetic Mean-Geometric Mean Inequality:
The arithmetic mean of a set of non-negative numbers is always greater than or equal to the geometric mean of the same set of numbers.
Proof:
Let's consider n positive numbers a₁, a₂, ..., aₙ whose product is unity.
We can write their arithmetic mean as:
AM = (a₁ + a₂ + ... + aₙ) / n
And their geometric mean as:
GM = (a₁ * a₂ * ... * aₙ)^(1/n)
Since the product of the numbers is unity, we have:
(a₁ * a₂ * ... * aₙ) = 1
Taking the n-th power on both sides, we get:
(a₁ * a₂ * ... * aₙ)^(1/n) = 1^(1/n)
GM = 1
Using the arithmetic mean-geometric mean inequality, we can write:
AM ≥ GM
Substituting the values of AM and GM, we get:
(a₁ + a₂ + ... + aₙ) / n ≥ 1
Multiplying both sides of the inequality by n, we have:
a₁ + a₂ + ... + aₙ ≥ n
Hence, the sum of n positive numbers whose product is unity is never less than n.
Conclusion:
Therefore, the correct answer is option 'D', the sum of n positive numbers is never less than n.
Arithmetic Mean-Geometric Mean Inequality:
The arithmetic mean of a set of non-negative numbers is always greater than or equal to the geometric mean of the same set of numbers.
Proof:
Let's consider n positive numbers a₁, a₂, ..., aₙ whose product is unity.
We can write their arithmetic mean as:
AM = (a₁ + a₂ + ... + aₙ) / n
And their geometric mean as:
GM = (a₁ * a₂ * ... * aₙ)^(1/n)
Since the product of the numbers is unity, we have:
(a₁ * a₂ * ... * aₙ) = 1
Taking the n-th power on both sides, we get:
(a₁ * a₂ * ... * aₙ)^(1/n) = 1^(1/n)
GM = 1
Using the arithmetic mean-geometric mean inequality, we can write:
AM ≥ GM
Substituting the values of AM and GM, we get:
(a₁ + a₂ + ... + aₙ) / n ≥ 1
Multiplying both sides of the inequality by n, we have:
a₁ + a₂ + ... + aₙ ≥ n
Hence, the sum of n positive numbers whose product is unity is never less than n.
Conclusion:
Therefore, the correct answer is option 'D', the sum of n positive numbers is never less than n.
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The product of n positive numbers is unity, then their sum is :a)a positive integerb)divisible by nc)equal ton + 1/nd)never less than nCorrect answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The product of n positive numbers is unity, then their sum is :a)a positive integerb)divisible by nc)equal ton + 1/nd)never less than nCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The product of n positive numbers is unity, then their sum is :a)a positive integerb)divisible by nc)equal ton + 1/nd)never less than nCorrect answer is option 'D'. Can you explain this answer?.
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