Question

The sum of all the odd numbers of four digit which are divisible by $ 9 $ , is:
A. 2754,000
B.2753.000
C.2752,000
D. none of these

Answer 1

Hint: To solve this question we will use the divisibility rule for 9 that is the sum of all the digits is divisible by 9. Then we write all the four-digit numbers which are divisible by 9. These numbers will be arithmetic progression. Hence, we will use the concept of sum of arithmetic progression to find the sum of all the numbers divisible by 9.

Complete step-by-step answer:
We know that the four-digit odd numbers that are divisible by 9 can be written as 1017, 1035,…….,9999.
We can say this can be expressed in the form of arithmetic progression AP.
The first term of arithmetic progression is $ a = 1017 $ .
The common difference of the arithmetic progression is $ d = 18 $ .
The last term of the arithmetic progression is $ L = 9999 $ .
Now we will find the number of terms in the arithmetic progression by using the expression for the last term of AP.
 $ L = a + \left( {n - 1} \right)d $
We will substitute 9999 for $ L $ , 1017 for $ a $ , 18 for $ d $ in the above expression to find the number of terms.
 $ \begin{array}{c}
9999 = 1017 + \left( {n - 1} \right)18\\
9999 - 1017 = 18n - 18\\
9999 - 1017 + 18 = 18n
\end{array} $
On rearranging the above expression, we get the value as,
 $ \begin{array}{c}
18n = 9000\\
n = \dfrac{{9000}}{{18}}\\
n = 500
\end{array} $
Now, we will find the sum of the arithmetic progression by the expression
 $ {S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right] $

We will substitute 500 for $ n $ , 1017 for $ a $ , 18 for $ d $ in the above expression to get the sum of all the four digits which are divisible by 9.
 $ \begin{array}{l}
{S_n} = \dfrac{{500}}{2}\left[ {2\left( {1017} \right) + \left( {500 - 1} \right) \times 18} \right]\\
{S_n} = 250\left( {2034 + 8982} \right)\\
{S_n} = 2754,000
\end{array} $
So, the correct answer is “Option A”.

Note: Make sure to find the correct value of the first and last term of arithmetic progression, since the number should be divisible by 9. Also in the formula of summation of arithmetic progression correctly put the values of all the terms to get the final answer.