the 3rd term of a geometric progression is nine times the 1st term.if the 2nd term is one-twenty fourth the 5th term.find the 4th term.(my solution so far).ar^2=9a r=sqr of 9 r=3.help me find the 1st term

a ar^1 ar^2 ar^3 ----- ar^(n-1)

ar^2 = 9a
so a = 3

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

r = (1/24) r^4

r^3 = 24 = 8*3 remember this
so
r = 2 *3^(1/3)

fourth = a r^3
= 3 *24 = 72

To find the 1st term (denoted as 'a') in a geometric progression, we can use the information given in the problem.

We know that the 3rd term (denoted as 't3') is nine times the 1st term (a). So, we can write this as:

t3 = 9a

Next, we are given that the 2nd term (denoted as 't2') is one-twenty fourth (1/24) the 5th term (denoted as 't5'). Mathematically, this can be expressed as:

t2 = (1/24) * t5

Now, let's consider the general formula for calculating terms in a geometric progression:

tn = a * r^(n-1)

Here, 'tn' represents the 'n'th term in the progression, 'a' is the 1st term, and 'r' is the common ratio.

We can use this formula to find the 2nd term and the 5th term. Let's substitute the values into the formula:

t2 = a * r^(2-1)
t5 = a * r^(5-1)

Now, we can substitute these expressions into the equation for t2 in terms of t5:

(1/24) * (a * r^(5-1)) = a * r^(2-1)

Simplifying this equation, we can cancel out 'a' from both sides:

(1/24) * r^4 = r

Multiplying both sides by 24 to eliminate the fraction gives:

r^4 = 24r

Now, we can rearrange the equation and factor out 'r':

r^4 - 24r = 0
r(r^3 - 24) = 0

Now, we have two possibilities:

1) r = 0 (which is not considered in our case since we are dealing with a non-zero progression)
2) r^3 - 24 = 0

To solve the second possibility, we can set the equation equal to zero and factorize:

r^3 - 24 = 0
(r - 2)(r^2 + 2r + 12) = 0

From this equation, we can see the value of 'r' is 2.

Therefore, we have determined that the common ratio 'r' is 3, and it is not 2.

Now, let's go back to the equation for t3:

t3 = 9a

We can substitute the value of 'r':

t3 = 9a = a * r^(3-1)
9a = a * 3^2
9a = 9a

From this equation, we see that 'a' could take any non-zero value, since both sides of the equation will always be equal.

Therefore, the 1st term 'a' can be any non-zero value.

To summarize, we have found that the 1st term 'a' can be any non-zero value, and the common ratio 'r' is 3.