Question:
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Solution:
The sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7 are:
103, 119...791
Here, we have:
a = 103
d = 16
$a_{n}=791$
We know:
$a_{n}=a+(n-1) d$
$\Rightarrow 791=103+(n-1) \times 16$
$\Rightarrow 688=16 n-16$
$\Rightarrow 704=16 n$
$\Rightarrow 44=n$
Also, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$
$\Rightarrow S_{44}=\frac{44}{2}[2 \times 103+(44-1) \times 16]$
$\Rightarrow S_{44}=22[206+688]$
$\Rightarrow S_{44}=22 \times 894=19668$