Question

Use the geometric sequence to help write a recursive rule and an explicit rule for any geometric sequence. for general rules, the values are of n and consecutive integers starting with 1.

Answer 1

Question: Use the geometric sequence to help write a recursive rule and an explicit rule for any geometric sequence:

Solution:

A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant. Remember that the constant factor (ratio) between consecutive terms of a geometric sequence is called the common ratio. Now, given a geometric sequence with the first term a1 and the common ratio r, the nth (or general) term is given by:

`a_n\text{ = }a_1r^(n-1)`

Thus, in this case, the common ratio r is:

`r\text{ = }(10)/(5)\text{ = 2}`

On the other hand, the first term of the sequence is a1 = 5. Then, we can conclude that the explicit rule for the given sequence is:

`a_{n\text{ }}=5(2)^(n-1)`

this is equivalent to:

`a_{n\text{ }}=5.2^n.2^(-1)`

this is equivalent to:

`a_{n\text{ }}=(5)/(2).2^n`

Then, the correct answer is:

The general rule:

`a_{n\text{ }}=(5)/(2).2^n`

the recursive rule:

`a_1=\text{ 5}`

`a_{n\text{ }}=(5)/(2).2^n`