A geometric progression is such that the 3rd term is nine times the 1st term while the 2nd term one twenty fourth of the 5th term.find its 4th term
"the 3rd term is nine times the 1st term" ---->ar^2 = 9a
r^2 = 9
r = ± 3
"the 2nd term one twenty fourth of the 5th term" ---> ar = (1/24)ar^4
24ar = ar^4
24 = r^3 <----- This contradicts the first condition in your question.
Fix your question.
To find the 4th term of the geometric progression, we need to determine the common ratio first. We can set up equations using the given information.
Let's assume that the first term of the geometric progression is "a" and the common ratio is "r".
From the given information, we have two equations:
1) The 3rd term is nine times the 1st term:
a * r^2 = 9a
2) The 2nd term is one twenty fourth of the 5th term:
a * r = (1/24) * a * r^4
We can simplify these equations:
1) r^2 = 9 (dividing both sides by 'a')
2) r^3 = 1/24 (dividing both sides by 'a' and multiplying by r)
To find the common ratio (r), we can take the cube root of both sides of equation 2:
r = (1/24)^(1/3)
r = 1/2
Now that we have the common ratio (r = 1/2), we can find the 4th term using the formula for the nth term of a geometric progression:
Tn = a * r^(n-1)
Plugging in the values, we have:
T4 = a * (1/2)^(4-1)
T4 = a * (1/2)^3
T4 = a * (1/8)
Therefore, the 4th term of the geometric progression is one-eighth (1/8) of the first term.