Question:
The product of n positive numbers is unity, then their sum is
The product of n positive numbers is unity, then their sum is
Updated On: Jun 23, 2023
- a positive integer
- divisible by n
- equal to $n+\frac{1}{n}$
- never less than n
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The Correct Option is D
Solution and Explanation
Since, product of n positive numbers is unity.
$\hspace30mm x_1. x_2. x_3 ...x_n = 1 \hspace30mm ...(i)$
$Using\, AM \ge GM, \frac{x_1+ x_2+ ...+x_n}{n} \ge (x_1. x_2....x_n)^{1/n} $
$\hspace30mm x_1+ x_2+ ... +x_n \ge n\, (1)^{1/n}\hspace5mm $ [from E (i)]
Hence, sum of n positive numbers is never less than n.
$\hspace30mm x_1. x_2. x_3 ...x_n = 1 \hspace30mm ...(i)$
$Using\, AM \ge GM, \frac{x_1+ x_2+ ...+x_n}{n} \ge (x_1. x_2....x_n)^{1/n} $
$\hspace30mm x_1+ x_2+ ... +x_n \ge n\, (1)^{1/n}\hspace5mm $ [from E (i)]
Hence, sum of n positive numbers is never less than n.
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Concepts Used:
Arithmetic Progression
Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.
For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.
In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.
For eg:- 4,6,8,10,12,14,16
We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.
Read More: Sum of First N Terms of an AP