Question

How many numbers from $1001$ to $2000$ are divisible by 4?

Answer 1

Hint: Write the first term by finding out the number which is divisible by 4 after 1000, that is 1004, similarly make an A.P. series by adding the common difference as 4.

Complete step by step solution:
Given, First number is 1001 and the last number is 2000, and the dividend is 4.
First number which is divisible from $1001$ to $2000$ is $1004$ and the last number is $2000$.
Common difference is 4.
Now, add $4$ in $1004$ and so on, a series will be formed which is
$ = 1004 + 1008 + 1012 + \ldots \ldots 2000$
It is observed that an Arithmetic Progression (A.P.) series is formed.
It is observed that there is also an last term in the series, use the formula of AP in this for last term to find the number of term in the series, that is ${{\text{T}}_n} = a + \left( {n - 1} \right)d$
Where, $a$ is the first term, $d$ is the common difference, ${{\text{T}}_n}$ is the last term and $n$ is the number of terms.
Substitute 1004 for $a$, 4 for $d$ and 2000 for ${{\text{T}}_n}$ in the formula ${{\text{T}}_n} = a + \left( {n - 1} \right)d$.
$2000 = 1004 + \left( {n - 1} \right)4$
Simplify for the number of terms, that is $n$.
$
  2000 = 1004 + 4n - 4 \\
  1000 = 4n \\
  250 = n \\
 $
Hence, the total number from$1001$ to $2000$ are divisible by $4$ are $250$.

Note: Use formula ${{\text{T}}_n} = a + \left( {n - 1} \right)d$, Do not use the formula for sum of $n$ terms of an arithmetic series, as last term is also given in the series, and do not add 4 as common difference in 1001, if we add it then it will becomes 1005, then series becomes wrong and 4 also not divide the number and series becomes wrong, then number of terms may also be wrong after finding.