In a G.P, the product of the second term and fourth term is double the fifth term and the sum of the first four terms is 80. Find the G.P.
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To find the common ratio and the first term of a geometric progression (G.P.), we need to solve the given conditions. Let's denote the common ratio as 'r' and the first term as 'a'.
According to the condition "the product of the second term and fourth term is double the fifth term," we can write the equation:
(a * r) * (a * r^3) = 2 * (a * r^4)
Simplifying this equation:
a^2 * r^4 = 2 * a * r^4
Dividing both sides by a, and then dividing both sides by r^4:
a = 2
Now, let's use the condition "the sum of the first four terms is 80." In a geometric progression, the sum of the first four terms can be obtained using the formula:
Sum = (a * (1 - r^n)) / (1 - r)
Here, we know the sum is 80, a is 2, and n is 4. Substituting these values into the formula, we have:
80 = (2 * (1 - r^4)) / (1 - r)
Simplifying the equation:
80(1 - r) = 2(1 - r^4)
80 - 80r = 2 - 2r^4
Rearranging the equation:
2r^4 - 80r + 78 = 0
Now we have a 4th degree polynomial equation. Solving it will give us the values of 'r'. However, this equation can be quite complex to solve algebraically. Alternatively, we can use numerical methods or approximation techniques to find the value(s) of 'r'.
(ar)(ar^3) = 2ar^4
a^2r^4 = 2ar^4
ar^4 = 2r^4
a = 2
a(r^4-1)/(r-1) = 80
1+r+r^2+r^3 = 40
r = 3
So, the GP is 2,6,18,54,162,...
(6)(54) = 2*162
2+6+18+54 = 80