30 points
can someone explain in detail how to solve the problem

Well

aritmetic is
an=a1+d(n-1)
so

first will be a1
7th is a1+6d

4th is a1+3d

let's say that a1=x and d=y for niceness sake

so

sum of first and 7th is 40
x+x+6y=40
2x+6y=40
divide 2
x+3y=20

product of first and 4th is 160
x times (x+3y)=160
hold  a sec, from previous problem, x+3y=20
x times 20=160
divide both sides by 20
x=8

sub back

x+3y=20
8+3y=20
minus 8 both sides
3y=12
divide by 3
y=4

an=8+4(n-1) is da equation
to find the terms, subsitute
frst is 8
2nd is 12
etc
the first 8 terms are
8,12,16,20,24,28,32,36

We know that,

First term = a; T7 = a + 6d; T4 = a + 3d

According to question,

a + a + 6d = 40

2a + 6d = 402(a + 3d) = 40a + 3d = 20 ...(1)

Now,

(a)(a + 3d) = 160

20a = 160a = 8 ...(2)

Plug the value of (2) in (1), we get

80 + 3d = 203d = 20 - 803d = - 60d = - 2.

So, AP => 8, 6, 4, 2, 0, - 2, - 4, - 6...