Question

the difference between the eighth and fourth of an AP is 40, if the eighth term is 1½ times the forth time find the common difference and the first of the AP

Answer 1

e = f + 4 d ... 4 d = 40

3/2 f = f + 40 ... 1/2 f = 40

1st = f - 3 d

Answer 2

a+7d - (a+3d) = 40
4d = 40
d = 10

a+7d = (3/2)(a+3d)
2a + 14d = 3a + 9d
5d = a
a = 50

check: term(8) = 50 + 70 = 120
term(4) = 50 + 30 = 80
the difference is 120-80 = 40 , check!
Is 120 = 1½ of 80 ? Yes, check!

btw, since it said the difference between term(8) and term(4) is 40
it could have been term4 - term8 = 40
in that case d = -10 and a = -50
it also checks.