Question:
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
Solution:
The two-digit numbers, which when divided by 4, yield 1 as remainder, are
13, 17, … 97.
This series forms an A.P. with first term 13 and common difference 4.
Let n be the number of terms of the A.P.
It is known that the $n^{\text {th }}$ term of an A.P. is given by, $a_{n}=a+(n-1) d$
$\therefore 97=13+(n-1)(4)$
$\Rightarrow 4(n-1)=84$
$\Rightarrow n-1=21$
$\Rightarrow n=22$
Sum of n terms of an A.P. is given by,
$\mathrm{S}_{n}=\frac{n}{2}[2 a+(n-1) d]$
$\therefore \mathrm{S}_{22}=\frac{22}{2}[22(13)+(22-1)(4)]$
$=11[26+84]$
$=1210$
Thus, the required sum is 1210.