Question

Find the sum of all two digit number which when divides by 3 leaves 1 as reminder

Answer 1

the numbers in the question can be defined by the expression 3x + 1.

the first term of this series is 10 & the last term is 97.

we now have this series:
10 + 13 + 16 + 19 + 22 + ... + 97
with a common diference, d = 3

this is an arithmetic series defined by;
a_n = a + ( n - 1 ) d

97 = 10 + ( n - 1 ) ( 3 )
.... = 10 + 3n - 3
.... = 3n + 7
3n = 90
n = 30

the sum of an arithmetic series is defined by;
S_n = ( n / 2 ) ( a_1 + a_n )

= ( 30 / 2 ) ( 10 + 97 )
= ( 15 ) ( 107 )
= 1,605