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

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

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

1st = f - 3 d

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.

To find the common difference and the first term of an arithmetic progression (AP), we can use the given information that the difference between the eighth and fourth terms is 40, and that the eighth term is 1½ times the fourth term.

Let's denote the first term of the AP as 'a', and the common difference as 'd'.

The formula for the nth term of an AP is given by:
An = a + (n-1)d, where An represents the nth term.

Given that the difference between the eighth and fourth terms is 40, we have:
A8 - A4 = 40

Substituting the formulas for the eighth and fourth terms, we get:
(a + 7d) - (a + 3d) = 40

Simplifying the equation, we have:
4d = 40

Dividing both sides by 4, we find that the common difference 'd' is 10.

Since we know that the eighth term (A8) is 1½ times the fourth term (A4), we can write:
A8 = 1.5 * A4

Substituting the formulas for A8 and A4, we have:
(a + 7d) = 1.5 * (a + 3d)

Expanding the equation, we get:
a + 7d = 1.5a + 4.5d

Rearranging the terms, we have:
7d - 4.5d = 1.5a - a

Simplifying the equation, we get:
2.5d = 0.5a

Dividing both sides by 2.5, we obtain:
d = 0.2a

Now, we have two equations:
d = 10
d = 0.2a

Substituting the value of d from the first equation into the second equation, we get:
10 = 0.2a

Dividing both sides by 0.2, we find that the first term 'a' is 50.

Therefore, the common difference of the AP is 10, and the first term is 50.