Question

does anyone know how to do geometric sequences with explicit formulas & recrusive formulas? i dont get it at all ._.

Answer 1

a recursive formula would be:
\[a_{n+1} = a_nr\]
where r is the common ratio between the terms. For example:
1, 2, 4, 8, 16, 32,....
In this sequence, the common ratio is 2, and to get the next term in the sequence, we multiply the current term by 2.

Answer 2

For a closed formula, you can figure it out as such. Lets say i have an initial term and a common ratio:
\[a_0, r\]
then the next term would be:
\[a_1 = a_0r\]
and the next term would be:
\[a_2 = a_1r = a_0r^2\]
and the next:
\[a_3 = a_2r = a_1r^2 = a_0r^3\]
and so on, so a closed formula would be:
\[a_n = a_0r^n\]
Now be a little careful. Since we started the sequence at the "zero-th" term, if the question asks for the 6th term, that means you need:
\[a_5 = a_0r^5\]

Answer 3

but how do you know which numbers to plug into where?

Answer 4

you always what to know what your first term and your common ratio is. from there you can figure anything out.
For example:
7, 14, 28, 56, 112,....
First i ask, "What is the first term?"
Thats kinda easy, its 7.
Then I ask, "what is the common ratio?"
To figure out, just divide two terms next to each other. You could do 14/7 = 2, or 112/56 = 2. It doesnt matter. So now you have your common ratio. Now you can answer anything.
What is the recursive formula for this sequence?
\[a_{n+1} = 2a_n\]
What is the closed forumla for this sequence?
\[a_n = a_0r^n \Rightarrow a_n = 7(2)^n\]
What is the 7th term?
\[a_6 = 7(2)^6 \Rightarrow a_6 = 7*64 \Rightarrow a_6 = 448\]
Remember, because i start the sequence at 0, the 7th term is a6, NOT a7