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Chapter 18: Problem 717

If mean of first \(n\) odd natural Integer is \(n\) then \(n\) is (a) 2 (b) 3 (c) 1 (d) any natural integer

Short Answer

Expert verified
The correct answer is \( (d) \) any natural integer.

Step by step solution

01

Write down the first n odd natural integers

To find the first n odd natural integers, we can represent them as: 1, 3, 5, ... , (2n - 1)
02

Calculate the sum of the first n odd natural integers

The sum of an arithmetic series is given by: Sum = (number of terms / 2) × (first term + last term) In our case, the number of terms = n, first term = 1, last term = 2n - 1. So, Sum = (n / 2) × (1 + (2n - 1)) Simplifying the equation, Sum = (n / 2) × (2n) Sum = n^2
03

Calculate the mean of the first n odd natural integers

The mean is given by the sum of the first n odd natural integers divided by the number of integers, which is n. Mean = (Sum of first n odd natural integers) / n Mean = (n^2) / n Mean = n
04

Compare the obtained mean with the given possible values of n

We have obtained the mean as n. Now we will check if the mean is equal to any of the given options. Option (a): 2 Option (b): 3 Option (c): 1 Option (d): any natural integer We know that the mean is equal to the value of n itself. This means that any natural integer could be a possible value for n.
05

Choose the correct option

Since the mean of the first n odd natural integers is equal to any natural integer value of n, the correct answer is: Option (d): any natural integer

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Most popular questions from this chapter

Given the observation \(5,9,13,17,25\) the mean deviation about the median is (a) \(5.5\) (b) \(5.8\) (c) 13 (d) \(5.6\)

The mean of the numbers \(a, b, 8,5,10\) is 6 and the variance is \(6.80\) then which one of following gives possible values of a and \(b\) ? (a) \(a=3, b=4\) (b) \(a=0, b=7\) (c) \(a=5, b=2\) (d) \(a=1, b=6\)

If the mean of numbers \(20+x, 24+x, 82+x, 100+x\) \(149+x\) is 75 then the mean of \(130+x, 126+x, 68+x\) \(50+x\) and \(1+x\) is (a) 75 (b) 76 (c) 73 (d) 70

If the mean and standard deviation of \(\mathrm{x}\) is \(\mathrm{b}\) and a respectively then the standard deviation of \([(x-b) / a]\) is (a) 1 (b) \((\mathrm{a} / \mathrm{b})\) (c) (b/a) (d) \(a b\)

If the mean deviation about the median of the observations a, \(2 a, \ldots \ldots . .50 a\) is 50 then \(|a|=\) (a) 2 (b) 3 (c) 4 (d) 5
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