The 4th term of an exponential sequence is 192 and 9th term is 6. Find the commonratio

To find the common ratio of an exponential sequence, we can use the formula for the nth term of an exponential sequence, which is:

a_n = a_1 * r^(n-1)

where:

• a_n is the nth term
• a_1 is the first term
• r is the common ratio
• n is the term number

We know that the 4th term (a_4) is 192 and the 9th term (a_9) is 6. We can set up two equations:

192 = a_1 * r^(4-1) => 192 = a_1 * r^3 ----(1)

6 = a_1 * r^(9-1) => 6 = a_1 * r^8 ----(2)

We can solve these two equations simultaneously to find the values of a_1 and r.

Divide equation (2) by equation (1):

(6/192) = (a_1 * r^8) / (a_1 * r^3)

0.03125 = r^5

Take the 5th root of both sides to solve for r:

r = (0.03125)^(1/5)

r = 0.5

So, the common ratio of the exponential sequence is 0.5.