Question

What is the term-to-term rule to this sequence? 64, 32, 16, 8, 4

Answer 1

`a_n = 1/2a_(n-1)`

Explanation:

`64, 32, 16, 8, 4`
is a geometric sequence.

this is a sequence where, to get from one term to the next, there is a number that you have to multiply it by. this is the same for all parts of the sequence.

`64/2 = 32`
`32/2 = 16`
etc.

to get from `64` to `32`, you divide by `2`.
this is the same as multiplying by `1/2`

`64 * 1/2 = 64/1 * 1/2 = 64/2 = 32`

hence, the number that each term is multiplied by to find the next is `1/2`.

if we call one term `a_n` and the previous term `a_(n-1)`, where `n` shows position in the sequence and therefore that `a_n` is one ahead of `a_(n-1)`, then we find that `a_n = 1/2a_(n-1)`.

this is the notation for the term-to-term rule, using `n` for the current position, `n-1` for the position before, and `a_n` with `a_(n-1)` to show progress from one position to the next.