The fourth term of an exponential sequence is 192 and its ninth term is 6. Find the common ratio of the sequence

fourth = 192

fifth = 192*r
sixth 192*r^2
seventh 192* r^3
eighth 192*r^4
ninth 192* r^5 =6
r^5 = 1/32
well I know 2^4 = 16 so 2^5 is 32
r = 1/2

To find the common ratio of an exponential sequence, we need to use the formula for the nth term of the sequence.

The formula for an exponential sequence is given by: aₙ = a₁ * r^(n-1)

Where:
aₙ = nth term of the sequence
a₁ = first term of the sequence
r = common ratio of the sequence
n = term number

Let's use the given information to calculate the common ratio. We have two pieces of information: the fourth term (a₄) which is 192, and the ninth term (a₉) which is 6.

Step 1: Substitute the values for a₉ and a₄ into the formula.

a₉ = a₁ * r^(9-1)
6 = a₁ * r^8

a₄ = a₁ * r^(4-1)
192 = a₁ * r^3

Step 2: Divide both equations to eliminate a₁.

(6/192) = (a₁ * r^8) / (a₁ * r^3)
(1/32) = r^5

Step 3: Take the fifth root of both sides.

r = (1/32)^(1/5)

Step 4: Simplify the right side.

r = 1/2

Therefore, the common ratio of the exponential sequence is 1/2.

To find the common ratio of an exponential sequence, we can use the formula:

\[a_n = a_1 \cdot r^{n-1}\]

where \(a_n\) is the n-th term of the sequence, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term.

We are given that the fourth term is 192 and the ninth term is 6. Using the formula, we can set up the following equations:

For the fourth term:
\[192 = a_1 \cdot r^{4-1}\]
\[192 = a_1 \cdot r^3 \quad \text{(equation 1)}\]

For the ninth term:
\[6 = a_1 \cdot r^{9-1}\]
\[6 = a_1 \cdot r^8 \quad \text{(equation 2)}\]

To find the common ratio, we can divide Equation 2 by Equation 1:

\[\frac{6}{192} = \frac{a_1 \cdot r^8}{a_1 \cdot r^3}\]
\[\frac{1}{32} = r^5\]

Taking the fifth root of both sides, we get:

\[r = \sqrt[5]{\frac{1}{32}}\]

Simplifying the right side:

\[r = \sqrt[5]{\frac{1}{2^5}}\]
\[r = \sqrt[5]{\frac{1}{32}}\]
\[r = \frac{1}{2}\]

Therefore, the common ratio of the sequence is \(r = \frac{1}{2}\).