Question

Find an explicit rule for the geometric sequence using subscript notation. Use a calculator and round your answers to the nearest tenth if necessary.

The third term of the sequence is 150. The fifth term is 96.


The explicit rule for the geometric sequence is an =

(

)n − 1.

Answer 1

a_n = 234.38 * (0.8 ^ (n - 1))

Explanation:

We have that the geometric sequence has the form of:

a_n = a * (r ^ (n - 1))

Now, we know that the third term has a value of: 150

And that the fifth term has a value of: 96

We know that the third term has the following form:

a_3 = a * (r ^ 2) = 150

And the fifth term has the following form:

a_5 = a * (r ^ 4) = 96

Therefore we have equations, if we divide both equations that is to say a_3 / a_5, we have to:

a * (r ^ 2) / a * (r ^ 4) = 150/96

The term a is canceled and we are left with:

1 / r ^ 2 = 1.5625

solving for r, we are left with:

r = (1 / 1.5625) ^ (1/2)

r = 0.8

Now to know the value of a, we use the following equation:

a_3 = a * (r ^ 2) = 150

we replace r and we are left with:

a * (0.8 ^ 2) = 150

a = 150 / (0.8 ^ 2)

a = 234,375

Therefore, we know all the terms and we can complete the equation:

a_n = 234.38 * (0.8 ^ (n - 1))