the second term of a GP is 36 more than the first term. The difference between the 4th and the 3rd terms is 900. Calculate the common ratio and the first term of the GP
Please solve for me
To solve this problem, let's first define the terms of the geometric progression (GP).
Let the first term be 'a' and the common ratio be 'r'.
Given that the second term is 36 more than the first term, we can write the following equation:
a * r = a + 36 ---(Equation 1)
Given that the difference between the fourth and third terms is 900, we can write the following equation:
a * r^3 - a * r^2 = 900 ---(Equation 2)
Now, let's solve these equations to find the values of 'a' and 'r'.
From Equation 1, we can rearrange it as:
a * (r - 1) = 36
Dividing both sides by (r - 1), we get:
a = 36 / (r - 1) ---(Equation 3)
Now, substitute Equation 3 into Equation 2:
(36 / (r - 1)) * r^3 - (36 / (r - 1)) * r^2 = 900
Simplifying this equation gives:
36r^3 - 36r^2 = 900(r - 1)
Dividing both sides by 36 gives:
r^3 - r^2 = 25(r - 1)
Rearranging this equation gives:
r^3 - r^2 - 25r + 25 = 0 ---(Equation 4)
To solve this cubic equation, we can utilize numerical methods or a graphing calculator. However, in this case, let's utilize trial and error method to find the value of 'r'.
We can start by trying different values of 'r' until we find one that satisfies Equation 4.
Let's try 'r = 2':
(2)^3 - (2)^2 - 25(2) + 25 = 8 - 4 - 50 + 25 = -21
The value of 'r = 2' does not satisfy Equation 4.
Let's try 'r = 3':
(3)^3 - (3)^2 - 25(3) + 25 = 27 - 9 - 75 + 25 = -32
The value of 'r = 3' also does not satisfy Equation 4.
Let's try 'r = 4':
(4)^3 - (4)^2 - 25(4) + 25 = 64 - 16 - 100 + 25 = -27
The value of 'r = 4' does not satisfy Equation 4.
Let's try 'r = 5':
(5)^3 - (5)^2 - 25(5) + 25 = 125 - 25 - 125 + 25 = 0
The value of 'r = 5' satisfies Equation 4.
Now that we have found the value of 'r', which is 5, we can substitute it into Equation 3 to find the value of 'a':
a = 36 / (r - 1) = 36 / (5 - 1) = 36 / 4 = 9
Therefore, the common ratio (r) of the geometric progression is 5, and the first term (a) is 9.
a1, a1 + 36, a3, a3 + 900
(a1 + 36) / a1 = (a3 + 900) / a3
1 + (36 / a1) = 1 + (900 / a3)
a3 = r^2 * a1
36 = 900 / r^2 ... 6 r = 30