Question

How many three digit number are divisible by $7$?

Answer 1

Three Numbers which are divisible by $7$ are

$105,112,119,.....994$

which forms an A.P

first term of this A.P $a_1=105$

second term of this A.P is $a_2=112$

common difference of this A.P is

$d=a_2-a1=112-105=7$

nth term of this A.P is given by

$a_n=a_1+(n-1)d$

$⟹a_n=105+(n-1)7$

$⟹a_n=105+7n-7$

$⟹a_n=98+7n...\dots .eq\left(1\right)$

for finding number of terms(n) in this A.P put last term $a_n=994$ in eq(1)

$⟹994=98+7n$

$⟹7n=994-98$

$⟹7n=896$

$⟹n=\frac{896}{7}$

$⟹n=128$

hence, their are $128$ three digit numbers which are divisible by $7$