The 3rd term of a geometric progression is nine times the first term.if the second term is one-twenty fourth the 5th term.find the 4th term. solution ar^2=9a r=sqr of 9 r=3.pls help me on how to get the first term(a)

ar^2 = 9a

so, r = ±3

Now we have a problem. We know that r=3, so the 5th term must be 27 times the 2nd term, regardless of the value of a.

And, as you say, there is no way to determine a from the information given so far.

You sure it's not 24 less than the 5th term?

Pls show work in details.

To find the value of the first term in the geometric progression, we can use the information given in the problem.

Let's denote the first term as 'a'.

The problem states that the third term is nine times the first term:
Third term = 9a

It is also given that the second term is one-twenty fourth (1/24) of the fifth term:
Second term = (1/24) * Fifth term

Let's denote the fifth term as 'b':
Fifth term = b

According to the problem, the second term is one-twenty fourth of the fifth term:
Second term = (1/24) * b

Now, let's use the formulas for the geometric progression:

Third term = a * r^2 (1)
Second term = a * r (2)
Fifth term = a * r^4 (3)

From equation (2), we have:
a * r = (1/24) * b

From equation (3), we have:
b = a * r^4

Substituting the value of 'b' from equation (3) into equation (2), we get:
a * r = (1/24) * (a * r^4)

Simplifying, we have:
r^3 = 1/24

Taking the cube root of both sides, we get:
r = (1/24)^(1/3)

Solving for 'r', we find that 'r' is approximately 0.5.

Now that we have the value of 'r', we can substitute it back into equation (1) to find the value of 'a':
(a * 0.5^2) = 9a

Simplifying, we get:
(0.25) * a = 9a

Dividing by 'a' on both sides, we get:
0.25 = 9

Clearly, the equation being inconsistent, we cannot find a unique value for 'a' using the given information. Hence, it is not possible to determine the exact value of the first term (a) in this geometric progression.

To find the first term (a) of the geometric progression, we can use the information given in the problem.

Let's start with the second term. It is mentioned that the second term is one-twenty fourth the fifth term. Mathematically, we can write this as:

a * r = (1/24) * a * r^4

Simplifying this equation, we can cancel out a and r from both sides, resulting in:

r^3 = 1/24

Next, we are given that the third term is nine times the first term. We can express this mathematically as:

a * r^2 = 9 * a

Again, we can simplify this equation by canceling out a and solving for r:

r^2 = 9

Taking the square root of both sides, we find:

r = 3

Now that we know the value of r, we can substitute it back into our equation for r^3:

(3)^3 = 1/24

Simplifying, we get:

27 = 1/24

To find the value of a, we need to isolate it on one side of the equation. We can do this by cross-multiplying:

27 * 24 = 1

a = 27 * 24

Therefore, the value of the first term (a) in this geometric progression is 648.

Now, using this information, we can find the fourth term. The fourth term can be found by multiplying the first term (a) by the common ratio (r) raised to the power of (4-1):

Fourth term = a * r^(4-1)

Substituting the values we know, we get:

Fourth term = 648 * (3)^3

Calculating this expression, we find:

Fourth term = 648 * 27

Thus, the fourth term of the geometric progression is 17,496.