The fifth term of an exponential sequence GP is greater than the fourth term by 13 whole number 1over 2 and the fourth term is greater than the third term by 9.find the answer to the common ratio
"by 13 whole number 1over 2"
I will assume you meant 9 1/2 or 19/2
ar^4 - ar^3 = 19/2 ---> ar^3(r - 1) = 19/2
ar^3 - ar^2 = 9 ---> ar^2(r - 1) = 9
divide those last two equations
r = 19/2 รท 9 = 19/18
"13 whole number 1over 2"
means 13 1/2 to me. ymvm
Let's assume the first term of the geometric progression (GP) is "a", and the common ratio is "r".
According to the given information, the fourth term is greater than the third term by 9. This can be represented as:
a * r^3 + 9 = a * r^2
Simplifying this equation, we get:
r^2 - r^3 = 9 / a ---- (Equation 1)
Similarly, the fifth term is greater than the fourth term by 13 1/2. This can be represented as:
a * r^4 + (13 + 1/2) = a * r^3
Simplifying this equation, we get:
r^3 - r^4 = (2 * 13 + 1) / 2a
r^3 - r^4 = 27 / 2a ---- (Equation 2)
Now, let's solve these two equations simultaneously to find the value of the common ratio (r).
Multiplying Equation 1 by 2 and multiplying Equation 2 by a, we get:
2r^2 - 2r^3 = 18 / a
ar^3 - ar^4 = 27 / 2
Combining these equations, we have:
2r^2 - 2r^3 - ar^3 + ar^4 = 18 / a - 27 / 2
Simplifying further, we get:
2r^2 - 2r^3 - ar^3 + ar^4 = (36 - 27a) / 2a
Dividing both sides by r^3, we have:
2/r - 2 - a + ar = (36 - 27a) / 2a * r
Adding ar to both sides and multiplying both sides by 2a, we get:
4a - 4ar + 2ar^2 - 2a^2r + a^2r^2 = 36 - 27a
Rearranging the equation, we have:
4ar - 2ar^2 + 2a^2r - a^2r^2 = 36 - 27a - 4a
4ar - 2ar^2 + 2a^2r - a^2r^2 + 27a + 4a - 36 = 0
Now, you can solve this quadratic equation to find the values of r.
To find the common ratio of the geometric progression (GP), we need to use the given information about the terms.
Let's assume that the first term of the GP is "a" and the common ratio is "r."
The fourth term can be expressed as:
fourth term = a * r^3
Similarly, the fifth term can be expressed as:
fifth term = a * r^4
According to the given information, the fifth term is greater than the fourth term by "13 1/2," which can be written as 13.5. So, we have:
fifth term - fourth term = 13.5
(a * r^4) - (a * r^3) = 13.5
a * r^3 * (r - 1) = 13.5
Also, it is mentioned that the fourth term is greater than the third term by 9:
fourth term - third term = 9
(a * r^3) - (a * r^2) = 9
a * r^2 * (r - 1) = 9
Now, we have two equations:
a * r^3 * (r - 1) = 13.5 -- Equation 1
a * r^2 * (r - 1) = 9 -- Equation 2
Dividing Equation 1 by Equation 2 will help eliminate "a" and simplify the equation:
[(a * r^3 * (r - 1)) / (a * r^2 * (r - 1))] = [13.5 / 9]
r = 1.5
Therefore, the common ratio (r) of the geometric progression is 1.5.