Question

The mean of first n natural number is X . What is the mean of first n even natural numbers

Answer 1

Answer:

Mean of first  n even natural numbers is 2X

Step-by-step explanation:

Formula used:

$Mean=\frac{\Sigma{x}}{n}$

Mean of first 'n' natural numbers:

$Mean=\frac{sum\:of\;first\:n\:natural\:numbers}{n}$

$X=\frac{1+2+.........+n}{n}$

$X=\frac{\frac{n(n+1)}{2}}{n}$

$X=\frac{n+1}{2}$................(1)

Mean of first 'n' even natural numbers:

$Mean=\frac{sum\:of\;first\:n\:even\:natural\:numbers}{n}$

$=\frac{2+4+.........+2n}{n}$

$=\frac{2(1+2+.........+n)}{n}$

$=\frac{\frac{2n(n+1)}{2}}{n}$

$=n+1$

$=2X$ (by using (1))

Answer 2

first n natural numbers are ; 1, 2, 3, 4, ..... n.

we know, mean = sum of observations/total number of observations

X = (1 + 2 + 3 + 4 + ... + n)/n

X = {n(n + 1)/2}/n

X = (n + 1)/2 ........(1)

[ note : sum of n natural numbers = n(n + 1)/2]

again, first n even natural numbers ; 2, 4, 6, 8, .... n

sum of first n even natural number = 2 + 4 + 6 + 8 + .... + n terms

it is an arithmetic progression,

so, using formula, S = n/2 [a + tn]

here, tn = a + (n - 1)d

= 2 + (n - 1)2 = 2n

now, sum of first n even natural numbers = n/2 [2 + 2n] = n(n + 1)

so, mean of first n even natural numbers = n(n + 1)/n = (n + 1)

from equation (1),

mean of first n even natural numbers = 2X