What's the missing digit?

$1^{st}$ $method:$ This problem can be solved using simple arithmetic. We can add all the $9$ numbers and then find their mean. Their sum $=9+99+999+9999+99999+999999+9999999+99999999+999999999$ $=1111111101$ The mean of these numbers $=\frac{1111111101}{9}$ $=123456789$ Thus, the digit not present in the mean is $\boxed{0}$ . $$ $2^{nd}$ $method:$ The other method is simpler and logical. As all the numbers have only the number $9$ as their digits, in order to find the mean we can simply divide each number by $9$ and add them. So we get the mean as $\frac{9}{9}+\frac{99}{9}+\frac{999}{9}+\frac{9999}{9}+\frac{99999}{9}+\frac{999999}{9}+\frac{9999999}{9}+\frac{99999999}{9}+\frac{999999999}{9}$ $=1+11+111+1111+11111+111111+1111111+11111111+111111111$ $=123456789$ So,the digit missing in the mean is $\boxed{0}$ .

The mean is $\dfrac{9+99+999+\cdots+999999999}{9}=1+11+111+\cdots+111111111=123456789$ The missing digit is $\boxed{0}$ .

one can easily get the answer without even adding the numbers

9 + 99 + 999........999999999

is same as

10 - 1 + 100 - 1 +........10^9 - 1

simplifying it

(10 + 100 +.....10^9) - (1*9)

Using GP series formula ( i.e. Sum = a(1-r^n)/1-r,

we get 10 + 100 + 1000......10^9 = 1111111110

= 1111111110 - 9 = 1111111101

For mean, Mean = 1111111101/9

which is 123456789

Therefore, we can get the answer without actually adding it easily. 0 is, therefore, the answer.

the mean $M=\frac{9+99+999+.....+999999999}{9}=1+11+111+......+111111111=123456789$

so the missing number is $\boxed{0}$

common sense : a mean of this particular series cannot have a 0 , for example start with a smaller series [ which we can replicate into a larger series later , by induction] for a odd sequence 9,99,999 , mean is 369, similarly take even sequence, 9,99,999,9999... so a series ending in 9 nines [a odd sequence] also should not contain 0.

here, the mean can b written as.....

(9+99+999+9999+99999+999999+9999999+99999999+999999999)/9

= (1+11+111+1111+11111+111111+1111111+11111111+111111111)

=123456789

so here the missing digit is 0.

The mean of 9,99,999,............,999999999 will be same as (1+11+111+............+111111111) which is exactly equal to 123456789 and hence 0 is the missing digit.

The average of these numbers is (9 + 99 + 999 + 9999 + ... + 999999999) / 9. Distributing the division, we get 123456789, which does not have a 0.