Find the sum of all two digit natural numbers which . . .

Question : Find the sum of all two digit natural numbers which when divided by 3 yields 1 as the remainder.

Doubt by Pushkar

Solution :

Two digit natural numbers which when divided by 3 yield 1 as a remainder are

10, 13, 16, . . . .97

a = 10

d = a₂-a₁
d = 13-10
d = 3

a_n=97

a_n=a+(n-1)d
97=10+(n-1)3

97-10=(n-1)3

87/3 = (n-1)
29=n-1

29+1=n
n=30

S_n=n/2[a+a_n]
S₃₀=30/2[10+97]
S₃₀=15[107]
S₃₀=1605