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
If geometric, then
(2k-4)/(k-3) = (4k-3)/(2k-4)
(2k-4)^2 = (k-3)(4k-3)
4k^2 - 16k + 16 = 4k^2 -15k + 9
-k = -7
k = 7
so the first 3 terms are: 4, 10, 25
a = 4, r = 2.5
sum(8) = a(r^8 - 1)/(r-1)
= 4(2.5^8 - 1)/(2.5-1)
= 4(390625/256 - 1)/(3/2)
= 4(390369/256)(2/3)
=4(130123/128)
= 130123/32 or 4066.34375
the actual first 8 terms are:
4, 10, 25, 62.5, 156.25, 390.625, 976.5625, 2441.40625
sum = 4066.634375
Yeahh
k-3 = a
r a = 2k-4
r^2 a = 4k-3
-------------------
r= (4k-3)/(2k-4) = (2k-4)/(k-3)
so
(2k-4)^2= (k-3)(4k-3)
4 k^2 -16 k + 16=4 k^2 - 15k +9
-k = -7
k = 7
go back and find a and r
Sure, let's put on our clown noses and solve this problem!
To find the value of K, we can use the fact that in a geometric progression, each term is equal to the previous term multiplied by a common ratio. In this case, let's call the common ratio "r".
The first term is K-3, and the second term is 2K-4. So, we can set up the equation:
2K-4 = (K-3) * r
Now, let's look at the second term and the third term (2K-4 and 4K-3):
4K-3 = (2K-4) * r
Now we have two equations. We can solve this system of equations to find the value of K.
Start by simplifying the first equation:
2K-4 = Kr-3r
2K - Kr = 4 - 3r
Next, simplify the second equation:
4K-3 = 2Kr - 4r
4K - 2Kr = 3 - 4r
To get rid of the fractions, let's multiply both equations by 2:
4K - 2Kr = 3 - 4r
4K - 2Kr = 8 - 6r
Now we can set the two equations equal to each other and solve for K:
3 - 4r = 8 - 6r
2r = 5
r = 5/2
Now, substitute the value of r back into one of the original equations to solve for K. Let's use:
2K-4 = Kr-3r
2K - 4 = K(5/2) - 3(5/2)
2K - 4 = (5K/2) - (15/2)
Multiply through by 2 to eliminate fractions:
4K - 8 = 5K - 15
K = 7
So, K is equal to 7.
Now, to find the sum of the first 8 terms of the geometric progression, we can use the formula:
Sum = a(1 - r^n) / (1 - r)
Where "a" is the first term and "r" is the common ratio. In this case, a = K - 3 and r = 5/2.
Substituting these values into the formula:
Sum = (K - 3)(1 - (5/2)^8) / (1 - 5/2)
Substituting K = 7:
Sum = (7 - 3)(1 - (5/2)^8) / (1 - 5/2)
Sum = 4(1 - (5/2)^8) / (1 - 5/2)
Calculating the expression, you'll get the sum of the first 8 terms of the progression.
I hope this clowny explanation helps! Let me know if you need further assistance.
In a geometric progression, each term is obtained by multiplying the previous term by a constant factor called the common ratio. Let's find the common ratio of this progression.
The second term, 2K - 4, is obtained by multiplying the first term, K - 3, by the common ratio. So, we have:
(2K - 4) = (K - 3) * r
Simplifying this equation: 2K - 4 = Kr - 3r
Now let's find the common ratio for the third term:
(4K - 3) = (2K - 4) * r
Simplifying: 4K - 3 = 2Kr - 4r
Now we have a system of two equations:
1) 2K - 4 = Kr - 3r
2) 4K - 3 = 2Kr - 4r
To solve this system, let's multiply equation 1 by 2:
2(2K - 4) = 2(Kr - 3r)
Simplifying: 4K - 8 = 2Kr - 6r
Now we have:
3) 4K - 8 = 2Kr - 6r
4) 4K - 3 = 2Kr - 4r
Subtracting equation 4 from equation 3 will eliminate the K terms:
(4K - 8) - (4K - 3) = (2Kr - 6r) - (2Kr - 4r)
Simplifying: -5 = -2r
Dividing both sides by -2:
r = 5/2
Now that we have the common ratio, let's find the value of K. We can substitute the common ratio into either of the original equations.
Using equation 1:
2K - 4 = (K - 3) * (5/2)
Multiplying both sides by 2:
4K - 8 = 5K - 15
Moving all the terms to one side:
5K - 4K = 15 - 8
K = 7
Therefore, the value of K is 7.
To find the sum of the first 8 terms of the progression, we can use the formula for the sum of a geometric progression:
Sum = a * (1 - r^n) / (1 - r)
Where a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = K - 3, r = 5/2, and n = 8.
Plugging in the values:
Sum = (K - 3) * (1 - (5/2)^8) / (1 - 5/2)
Calculating the sum using K = 7:
Sum = (7 - 3) * (1 - (5/2)^8) / (1 - 5/2)
Sum = 4 * (1 - 390625/256) / (1/2)
Simplifying: Sum = 4 * (-390621/256) / (1/2)
Multiplying numerator and denominator by 2:
Sum = 4 * (-781242/512)
Simplifying: Sum = -3124968/512 = -6103.25
Therefore, the sum of the first 8 terms of the progression is -6103.25.
To find the value of K and the sum of the first 8 terms of the geometric progression, we can use the fact that a geometric progression follows a consistent pattern where each term is obtained by multiplying the previous term by a constant called the common ratio (r).
Step 1: Find the common ratio (r)
To find the common ratio, we can divide the second term by the first term:
(ratio between second term and first term) = (2K-4)/(K-3)
Similarly, we can divide the third term by the second term:
(ratio between third term and second term) = (4K-3)/(2K-4)
Since these ratios must be equal in a geometric progression, we can set them equal to each other and solve the equation:
(2K-4)/(K-3) = (4K-3)/(2K-4)
Step 2: Solve for K
To solve the equation, we can cross-multiply:
(2K-4)(2K-4) = (K-3)(4K-3)
Expanding both sides of the equation:
4K^2 - 16K + 16 = 4K^2 - 15K + 9
Simplifying the equation:
K = 7
Therefore, the value of K is 7.
Step 3: Find the common ratio (r)
Now that we know K is 7, we can substitute it back into one of the equations to find the common ratio (r):
(ratio between second term and first term) = (2(7)-4)/(7-3)
= 10/4
= 2.5
Therefore, the common ratio (r) is 2.5.
Step 4: Find the sum of the first 8 terms
To find the sum of the first 8 terms of the geometric progression, we can use the formula:
Sum = a(1 - r^n) / (1 - r)
Where:
a = first term of the geometric progression (K-3)
r = common ratio (2.5)
n = number of terms to be added (8)
Substituting the values:
Sum = (K-3)(1 - 2.5^8) / (1 - 2.5)
Sum = (7-3)*(1 - 2.5^8) / (1 - 2.5)
Simplifying the expression:
Sum = 5040
Therefore, the sum of the first 8 terms of the geometric progression is 5040.