Question

the mean of first n natural numbers is x what is the mean of first n even natural numbers​

Answer 1

Solution:

Given mean of first n natural numbers  is "x"

let "y" be sum of first n natural numbers  and "z" be the sum of first n even natural numbers.

we know according to formula of mean

 mean of first n  natural numbers = y / n    .......(1)

 mean of first   n   even  numbers = z / n    .......(2)

Since:

if we multiplay each natural number from the lest of given first n natural numbers by 2 we get the first n even numbers

                                Therefore

sum of first n even natural numbers= 2 * (sum of first n natural number )

sum of first n even natural numbers=2 *(y)=2y

                                    This implies

                                                z=2y

putting the z=2y in equation (2) we get

 mean of first   n   even  numbers = 2y / n=2(y/n)

 mean of first   n   even  numbers  =2(mean of first n  natural numbers)

Given mean of first   n   natural  numbers  = x

So

mean of first   n   even  numbers  =2(x)=2x.

Answer 2

Answer:

Mean of  First n Even natural numbers  = 2x

Step-by-step explanation:

First n natural numbers

1 , 2 , 3 , ........................... n

Sum =  1 + 2 + 3 + ............... + n

Mean = Sum/n

=> x = ( 1 + 2 + 3 + ............... + n)/n     Eq 1

First n Even natural numbers

2 , 4 , 6 , ........................... 2

Sum = 2 + 4 + 6 + ............... + 2n

Taking two common

Sum = 2 * ( 1 + 2 + 3 + ............... + n)

Mean = Sum/n = 2 * ( 1 + 2 + 3 + ............... + n)/n

Putting value of x from eq 1

Mean = 2 * x

Mean of  First n Even natural numbers  = 2x