In a gp the product of the 2nd and the 4th term is doubled the 5th term & the sum of the first four term is 80.find the gp
I will read that as:
the product of the 2nd and the 4th term is double the 5th term
---> (ar)(ar^3) = 2(ar^4)
a^2 r^4 = 2ar^4
divide by ar^4
a = 2
the sum of the first four term is 80
---> a(r^4 - 1)/(r - 1) = 80
but a = 2, so
r^4 - 1 = 40(r - 1)
r^4 - 40r + 39 = 0
r = 3, r = 1, or r = two complex roots
but in a GP, r ≠ 1
so r = 3
pressed <submit> too soon, but I am sure you can state the GP.
To find a geometric progression (GP) in this given scenario, we need to analyze the information given and use the formulas for GP.
Let's assume that the first term of the GP is 'a', and the common ratio is 'r'. Therefore, the second term is 'ar', the third term is 'ar^2', and so on.
Now, let's break down the given information step by step:
1. "The product of the 2nd and the 4th term is double the 5th term":
(ar)(ar^3) = 2(ar^4)
2. "The sum of the first four terms is 80":
a + ar + ar^2 + ar^3 = 80
Now, let's solve these equations:
1. (ar)(ar^3) = 2(ar^4)
ar^4 = 2ar^4
Divide both sides by ar^4 (assuming r is not equal to 0):
a = 2
2. a + ar + ar^2 + ar^3 = 80
Substitute the value of 'a' from the previous equation:
2 + 2r + 2r^2 + 2r^3 = 80
Divide both sides by 2:
1 + r + r^2 + r^3 = 40
At this point, we have a cubic equation. To find the value(s) of 'r', we can use numerical methods or solve it graphically. One way to solve it numerically is by using a calculator or software that can solve polynomial equations.
The equation simplifies to:
r^3 + r^2 + r + 1 - 40 = 0
r^3 + r^2 + r - 39 = 0
Using a numerical method, like the Newton-Raphson method, we can find that one of the real solutions to this equation is r = 3.
Finally, substituting the value of 'r' back into the equation a = 2, we find that the first term, a, is 2. Therefore, the geometric progression (GP) is:
2, 6, 18, 54, ... (with a common ratio of 3)