Question: The arithmetic mean of the nine numbers in the given set {9,99,999, ........999999999} is a 9-digit ...

Question

The arithmetic mean of the nine numbers in the given set {9,99,999, ........999999999} is a 9-digit number N, all whose digits are distinct, the number N does not contain the digit
A. 0
B. 2
C. 5
D. 9

Answer 1

Solution

Hint: Proceed the solution of this question, with the concept of arithmetic mean and try to visualise any pattern that we can form by taking something common like we can take 9 as a common in this particular question.

Complete step-by-step answer:
For the given question
We know that arithmetic mean is one of the most common ways to determine the measure of a central tendency.
Arithmetic mean N= $\dfrac{{\left( {{\text{sum of terms}}} \right)}}{{{\text{numbers of terms}}}}$
Here there are nine terms $9,99,999,9999,99999,...............999999999$
So their arithmetic mean N =
$\Rightarrow \dfrac{{\left( {9 + 99 + 999 + 9999 + 99999 + 999999 + 9999999 + 99999999 + 999999999} \right)}}{9}$
On taking 9 as common from numerator
$\Rightarrow \dfrac{{9\left( {1 + 11 + 111 + 1111 + 11111 + 111111 + 1111111 + 11111111 + 111111111} \right)}}{9}$
Cancel 9 from numerator and denominator both

$\Rightarrow 1 + 11 + 111 + 1111 + 11111 + 111111 + 1111111 + 11111111 + 111111111$

Now see this pattern
⇒ 1+11=12,
⇒1+11+111=123,
⇒1+11+111+1111=1234
⇒1+11+111+1111+11111=12345
⇒1+11+111+1111+11111+111111=123456
Similarly, this pattern will so on. And at 9th term will be same as equation (1) which is the expression of our required arithmetic mean N
.
.
.
⇒ 1+11+111+1111+11111+111111+1111111+11111111+111111111=123456789

Arithmetic Mean N =123456789.
Hence arithmetic mean N does not contain 0 as one of its digit
Hence option (A) is correct

Note: In such types of questions of sequence and series, the number of terms are more, most of the time it is not feasible to add the whole series manually and also it is not forming any particular series like AP, GP or HP so we can’t use any formula to find its sum. Hence in such a type of situation, we should try to visualise any pattern which might be repeating hence in this way we can find our required result.