In a Gp the product of the second and the fourth term is double the fifth term and the sum of the first four term is 80. Find the first five terms of the Gp

Using your definitions of a GP ....

"the product of the second and the fourth term is double the fifth term "
---> (ar)(ar^3) = 2(ar^4)
a^2 r^4 = 2a r^4
a = 2

"the sum of the first four term is 80"
---> a + ar + ar^2 + ar^3 = 80
2(1 + r + r^2 + r^3) = 80
1 + r + r^2 + r^3 - 40 = 0
r^3 + r^2 + r - 39 = 0
Try ±3 and ± 13
sure enough , r = 3 satisfies the equation
did a long division and got
r^3 + r^2 + r - 39 = (r-3)(r^2 + 4r + 13)
The quadratic does not have any real roots, so r = 3

take it from there.

explain

how did u do the long division please?

To find the first five terms of a geometric progression (GP) given certain conditions, we need to set up equations based on the information provided in the question.

Let's assume the first term of the GP is "a", and the common ratio is "r".

According to the given conditions:
1) The product of the second and the fourth terms is double the fifth term. This can be written as:

(ar)(ar^3) = 2(ar^4)

Simplifying this equation, we get: a^2r^4 = 2ar^4

Dividing both sides by r^4, we have: a^2 = 2r

2) The sum of the first four terms is 80. This can be written as:

a + (ar) + (ar^2) + (ar^3) = 80

Now, we have two equations with two variables (a and r). Let's solve them simultaneously to find their values.

Using equation (2), we can express "a" in terms of "r":
a = 80 - (ar) - (ar^2) - (ar^3) ...(3)

Now, substitute the value of "a" from equation (3) into equation (1):
(80 - (ar) - (ar^2) - (ar^3))^2 = 2r

Expanding and simplifying this equation will give us a quadratic equation in "r". Solving this quadratic equation will give us the values of "r".

Once we have the value of "r", we can substitute it back into equation (3) to find the value of "a".

Finally, by using these values of "a" and "r", we can find the first five terms of the GP by calculating:
First term = a
Second term = ar
Third term = ar^2
Fourth term = ar^3
Fifth term = ar^4