If the sum of the first two terms and the sum of the first four terms ...
To solve this problem, we need to use the given information about the geometric progression and find the sixth term.
Let's denote the first term of the geometric progression as 'a' and the common ratio as 'r'.
The sum of the first two terms can be written as:
a + ar = 8
The sum of the first four terms can be written as:
a + ar + ar^2 + ar^3 = 80
We can rewrite the second equation by factoring out 'a' and using the formula for the sum of a geometric series:
a(1 + r + r^2 + r^3) = 80
Simplifying further, we have:
a(1 - r^4)/(1 - r) = 80
Since the common ratio 'r' is positive, we know that 0 < r="" />< 1.="" this="" means="" that="" as="" 'r'="" increases="" to="" approach="" 1,="" the="" value="" of="" (1="" -="" r^4)/(1="" -="" r)="" also="" increases.="" therefore,="" to="" find="" the="" largest="" possible="" value="" of="" 'a',="" we="" need="" to="" find="" the="" value="" of="" 'r'="" that="" makes="" the="" numerator="" (1="" -="" r^4)="" as="" large="" as="" />
To do this, we can take the derivative of the numerator with respect to 'r' and set it equal to zero to find the critical points. Taking the derivative, we have:
d/dx (1 - r^4) = -4r^3
Setting this equal to zero, we get:
-4r^3 = 0
r = 0
However, since 'r' must be positive, we can discard this critical point. This means that the numerator (1 - r^4) does not have any critical points in the interval 0 < r="" />< 1.="" therefore,="" the="" maximum="" value="" of="" the="" numerator="" occurs="" at="" either="" r="0" or="" r="" />
Since the common ratio cannot be 0 (as it is given to be positive), the maximum value of the numerator occurs when r = 1. Plugging this value back into the equation, we have:
a(1 - 1^4)/(1 - 1) = 80
a(0/0) = 80
Since the numerator is 0, this equation is undefined. However, we know that the sum of the first four terms is 80, so the numerator must be equal to 0. This means that a = 0.
Therefore, the first term of the geometric progression is 0, and the common ratio is positive. We can now find the sixth term:
a = 0
r = 8/0 = undefined
Using the formula for the nth term of a geometric progression, we have:
a6 = a * r^5 = 0 * (8/0)^5
Since any number raised to the power of 0 is 1, this simplifies to:
a6 = 0 * 1 = 0
Therefore, the sixth term of the geometric progression is 0. However, none of the answer options provided match this result. It is possible that there is an error in the question or the answer options.