A G.p is such that the 3rd term is 9 times the first term, while the second term is 124 of the fifth. Find the fourth term
since T3 = T1 * r^2, r = 3
No idea what "124 of the fifth" means
To find the fourth term of the geometric progression (G.P.), we need to understand the pattern that defines a G.P.
A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio (r).
Let's assume the first term of the G.P. is "a" and the common ratio is "r".
Given:
The third term is 9 times the first term.
The second term is 1/24 of the fifth term.
From these conditions, we can set up two equations to represent the given information.
Equation 1: a * r^2 = 9a
Equation 2: a * r = (1/24) * a * r^4
Now, let's solve these equations to find the values of "a" and "r".
From Equation 1, we get: r^2 = 9
Taking the square root of both sides, we find: r = ±3
Since a G.P. cannot have a negative common ratio, we consider only the positive value, r = 3.
Now, let's substitute this value of "r" into Equation 2 to solve for "a".
a * 3 = (1/24) * a * 3^4
3a = (1/24) * 81a
72a = 81a
81a - 72a = 0
9a = 0
Thus, a = 0.
Now that we have found the value of "a" and "r", we can determine the fourth term by calculating a * r^3.
Fourth term = 0 * 3^3 = 0.
Therefore, the fourth term of the geometric progression is 0.
To find the fourth term of the geometric progression (G.P.), we need to know the common ratio (r) of the G.P.
Let's assume that the first term of the G.P. is "a" and the common ratio is "r".
Given information:
The third term is 9 times the first term:
a * r^2 = 9a --------------(1)
The second term is 1/24 of the fifth term:
a * r = (1/24)(a * r^4) --------------(2)
To solve for r, we can divide equation (2) by equation (1):
(a * r) / (a * r^2) = (1/24)(a * r^4) / 9a
r / r^2 = (r^4) / (24 * 9)
1 / r = r^2 / (24 * 9)
1 = r^3 / (24 * 9)
r^3 = 24 * 9
Now, let's solve for the value of r:
r^3 = 24 * 9
r^3 = 216
Taking the cube root of both sides:
r = ∛216
r = 6
Now, we have the value of r, we can substitute it into equation (1) to find the first term (a):
a * 6^2 = 9a
a * 36 = 9a
36a = 9a
27a = 0
a = 0
Therefore, the first term (a) of the G.P. is 0.
Now we can find the fourth term using the formula for the nth term of a G.P.:
Fourth Term = a * r^(n-1)
Fourth Term = 0 * 6^(4-1)
Fourth Term = 0 * 6^3
Fourth Term = 0
Therefore, the fourth term of the geometric progression is 0.