Question

An arithmetic progression is such that the 9th term is 0 and the sum of the first 25 terms is 50 so, how do I find the first term

Answer 1

just use your definitions:
term(9) = a + 8d = 0 , so a = -8d
sum(25) = (25/2)(2a + 24d) = 50
25(2a +24d) = 100
2a + 24d = 4
a + 12d = 2
-8d + 12d = 2
4d = 2
d = 1/2
then a = -8(1/2) = - 4 <----- your first term by definition of a