The first three terms of a G.P are (k-3) (2k – 4), (4k – 3) in that order. Find the value K and the sum of the first eight terms
since there is a common ratio r, you can find k by solving
(2k-4)/(k-3) = (4k-3)/(2k-4)
Now you can determine r, and
S8 = (k-3)(r^8-1)/(r-1)
To find the value of k, we can use the property of geometric progression (G.P) where the ratio between consecutive terms is constant.
Let's find the ratio (r) between the second and first terms:
r = (2k - 4) / (k - 3)
Now, let's find the ratio between the third and second terms:
r = (4k - 3) / (2k - 4)
Since both ratios are equal, we can equate them:
(2k - 4) / (k - 3) = (4k - 3) / (2k - 4)
Cross-multiplying and simplifying this equation gives us:
(2k - 4) * (2k - 4) = (k - 3) * (4k - 3)
Expanding and simplifying further:
4k^2 - 16k + 16 = 4k^2 - 15k + 9
Subtracting 4k^2 from both sides:
-16k + 16 = -15k + 9
Rearranging terms:
16k - 15k = 9 - 16
k = -7
Therefore, the value of k is -7.
To find the sum of the first eight terms, we can use the formula for the sum of a geometric progression:
S = a * (r^n - 1) / (r - 1)
Here, a is the first term, r is the common ratio, and n is the number of terms.
The first term, a, is (k - 3) (substituting k = -7):
a = (-7 - 3) = -10
The common ratio, r, can be found from the ratio between the second and first terms (which we calculated earlier):
r = (2k - 4) / (k - 3)
r = (2(-7) - 4) / (-7 - 3)
r = (-14 - 4) / (-10)
r = -18 / -10
r = 9 / 5
Using these values, we can find the sum of the first eight terms, S:
S = -10 * ( (9/5)^8 - 1 ) / (9/5 - 1)
Calculating this expression gives:
S ≈ -10 * (6.19174 - 1) / (4/5)
S ≈ -10 * (5.19174) / (4/5)
S ≈ -10 * 6.489675
S ≈ -64.89675
Therefore, the sum of the first eight terms is approximately -64.89675.
To find the value of k in the given geometric progression (G.P.), we'll use the fact that consecutive terms in a G.P. have a common ratio.
The common ratio (r) is found by dividing any term by its preceding term. In this case, we can calculate the common ratio by dividing the second term by the first term:
(2k - 4) ÷ (k - 3) = (4k - 3) ÷ (2k - 4)
Now, let's solve this equation to find the value of k:
(2k - 4) ÷ (k - 3) = (4k - 3) ÷ (2k - 4)
Cross-multiplying, we get:
(2k - 4) * (2k - 4) = (k - 3) * (4k - 3)
Simplifying both sides:
4k^2 - 16k + 16 = 4k^2 - 15k + 9
Subtracting 4k^2 from both sides:
-16k + 16 = -15k + 9
Rearranging and simplifying further, we have:
16 - 9 = -15k + 16k
7 = k
Therefore, k = 7.
To find the sum of the first eight terms of the G.P., we'll use the formula for the sum of a finite G.P., which is given by:
Sum(n) = (first term * (1 - common ratio^n)) / (1 - common ratio)
Here, the first term is (k - 3) (2k - 4), and the common ratio is the value of r we found earlier. Also, n is 8 since we want to find the sum of the first 8 terms.
Substituting these values into the formula:
Sum(8) = ((k - 3) (2k - 4) * (1 - r^8)) / (1 - r)
Let's calculate this sum using the value of k that we found:
Sum(8) = ((7 - 3) (2 * 7 - 4) * (1 - r^8)) / (1 - r)
Simplifying further:
Sum(8) = (4 * 10 * (1 - r^8)) / (1 - r)
Now, substitute the value of r into the equation:
Sum(8) = (4 * 10 * (1 - (2/7)^8)) / (1 - 2/7)
Calculating this expression will give us the sum of the first eight terms of the G.P.