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 170. The fifth term is 108.8.

Answer 1

To derive the explicit rule for the geometric sequence with given terms, calculate the common ratio using the provided terms and then find the first term. With these values, the rule is expressed as an = a1 × r(n-1), where a1 is the first term and r is the common ratio.

Step-by-step explanation:

To find an explicit rule for the geometric sequence given the third term (170) and the fifth term (108.8), we need to determine the common ratio (r) and the first term (a1). The ratio between consecutive terms is constant in a geometric sequence and can be calculated using the formula:

an = a1 × r(n-1)

Using the information for the third and fifth terms, we can set up the following equations:

170 = a1 × r(3-1)

108.8 = a1 × r(5-1)

Now, divide the second equation by the first to find the common ratio r:

\( \frac{108.8}{170} = r(5-3) \)

\( r = \sqrt{\frac{108.8}{170}} \)

Use a calculator to find the value of r and round to the nearest tenth if necessary.

Once you have r, you can substitute it back into one of the original equations to solve for a1. Then the explicit rule for the geometric sequence would be:

an = a1 × r(n-1)

Where a1 is the first term of the sequence and r is the common ratio.

Answer 2

The formula for nth term is given as
`a_n = (265.625)(0.8)^((n-1))`

Step-by-step explanation:

Here, 3rd term in the geometric sequence = 170

The 5th term in the sequence = 108.8

Let the first term in the sequence = a

Let the common ratio of the sequence = r

Now by the Geometric Sequence:

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

⇒ From above general term:

`a_3  = a r^(2) ,\\ a_5 = ar^(4)`

⇒
`170  = a r^(2) ,\\ 108.8 = ar^(4)`

Dividing both the the equations, we get:

`(170)/(108.8)  = (ar^2)/(ar^4)  \implies 1.5625 = (1)/(r^2)`

or,
`r^(2)   = (1)/(1.5625)  = 0.64\\ \implies r = 0.8`

Hence, the common ratio r = 0.8

Now,
`170 =  a r^2  \implies 170 = a (0.64)\\\implies a = (170)/(0.64)  = 265.625`

⇒ a = 265.625, r = 0.8

So, the formula for nth term is given as
`a_n = (265.625)(0.8)^((n-1))`