Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
The integers from 1 to 100, which are divisible by 2, are 2, 4, 6… 100.
This forms an A.P. with both the first term and common difference equal to 2.
⇒100 = 2 + (n –1) 2
⇒ n = 50
$\therefore 2+4+6+\ldots+100=\frac{50}{2}[2(2)+(50-1)(2)]$
$=\frac{50}{2}[4+98]$
$=(25)(102)$
$=2550$
The integers from 1 to 100, which are divisible by 5, are 5, 10… 100.
This forms an A.P. with both the first term and common difference equal to 5.
$\therefore 100=5+(n-1) 5$
$\Rightarrow 5 n=100$
$\Rightarrow n=20$
$\therefore 5+10+\ldots+100=\frac{20}{2}[2(5)+(20-1) 5]$
$=10[10+(19) 5]$
$=10[10+95]=10 \times 105$
$=1050$
The integers, which are divisible by both 2 and 5, are 10, 20, … 100.
This also forms an A.P. with both the first term and common difference equal to 10.
$\therefore 100=10+(n-1)(10)$
$\Rightarrow 100=10 n$
$\Rightarrow n=10$
$\therefore 10+20+\ldots+100=\frac{10}{2}[2(10)+(10-1)(10)]$
$=5[20+90]=5(110)=550$
$\therefore$ Required sum $=2550+1050-550=3050$
Thus, the sum of the integers from 1 to 100, which are divisible by 2 or 5, is 3050.