The two digit numbers when divided by 4, yields 1 as remainder are:

5, 9,13, 17,................,97

The given series is in A.P.

first term = a =5

common difference = d = 4

Total number of terms in the series 

l = a+ (n-1)*d   where n is the total number of terms in the series

97 = 5 + (n-1)*4

=>97-5 = (n-1)*4

=>92 =  (n-1)*4

=> n-1 = 92/4

=>n-1 = 23

=>n = 23+1

=>n = 24

Now sum of the series = (n/2)*(a+l)  (l is the last term of the series)

=>(24/2)*(5+97)

=>12*102

=>1224