The first three terms of a geometric progression are K-3,2K-4,4K-3 in that order find the value of K and the sum of the first 8 terms of the progression.
This is a strange question if the subject you are studying is "Hill school." I never heard of "Hill school" as a subject for study.
It's a question in mathematics under geometric progression
To find the value of K in the geometric progression, we can use the fact that consecutive terms in a geometric progression have a common ratio.
The common ratio (r) can be calculated by dividing any term in the progression by the previous term. Let's take the second term, (2K - 4), and divide it by the first term, (K - 3):
(2K - 4) / (K - 3)
Similarly, we divide the third term, (4K - 3), by the second term, (2K - 4):
(4K - 3) / (2K - 4)
Since both of these ratios represent the common ratio (r), we can equate them:
(2K - 4) / (K - 3) = (4K - 3) / (2K - 4)
To solve this equation, we can cross-multiply:
(2K - 4) * (2K - 4) = (K - 3) * (4K - 3)
Expanding both sides:
4K^2 - 16K + 16 = 4K^2 - 7K + 9
Simplifying:
-16K + 16 = -7K + 9
Bringing like terms to one side:
-16K + 7K = 9 - 16
Simplifying further:
-9K = -7
Dividing both sides by -9:
K = -7 / -9
K = 7/9
Therefore, the value of K is 7/9.
To find the sum of the first 8 terms of the geometric progression, we can use the formula for the sum of a finite geometric series:
Sum = a * (1 - r^n) / (1 - r)
Where:
a = first term of the series
r = common ratio of the series
n = number of terms in the series
In this case, the first term (a) is (K - 3), the common ratio (r) is ((2K - 4) / (K - 3)), and the number of terms (n) is 8.
Substituting these values into the formula:
Sum = (K - 3) * (1 - ((2K - 4) / (K - 3))^8) / (1 - ((2K - 4) / (K - 3)))
Now, substitute the value of K we found earlier, which is 7/9:
Sum = ((7/9) - 3) * (1 - ((2(7/9) - 4) / ((7/9) - 3))^8) / (1 - ((2(7/9) - 4) / ((7/9) - 3)))
Simplify the equation and calculate the sum to find the final answer.