Question

Is it arithmetic or geometric, and solve for n ( basically what is the formula and the sequence derived?)

256, 192, 144....n=7

Answer 1

It is geometric with common ratio 0.75 .

The explicit formula for the sequence is
`a_n=256 \cdot (0.75)^(n-1)`.

The 7th term is
`a_7` which is 45.5625 .

Explanation:

It is arithmetic if you can add/subtract the same number to get from term to the term after. In arithmetic sequence, the number you can add to get from term to term after is called the common difference. It is called common difference because it is the same number over and over.

It is geometric if term divided previous term is the same number each time. This is called a common ratio.

Let's test to see if it is arithmetic.

If it is arithmetic, all of the following will have the same difference:

192-256=?

144-192=?

----------------------

192-256=-64

144-192=-48

These aren't the same number. The sequence is not arithmetic.

Let's test to see if is geometric.

If it is geometric, all of the following will have the same ratio:

192/256=?

144/192=?

---------------------------

192/256=3/4 (or .75)

144/192=3/4 (or .75)

These are the same number. The sequence has a common ratio and is therefore geometric.

The explicit form for the equation of a geometric sequence is:

`a_n=a_1 \cdot r^(n-1)`

where
`a_1` is the first term and
`r` is the common ratio.

We have
`a_1=256` while
`r=.75`.

The explicit form is:

`a_n=256 \cdot (.75)^(n-1)`

The 7th term can be found by replacing
`n` with 7. Like so:

`a_7=256 \cdot (.75)^(7-1)`

Putting right hand side into calculator reveals the 7th term is:

`a_7=45.5625`

Answer 2

It is geometric because it is a sequence of multiplying a constant number, in this case the constant is 0.75. You find the constant by dividing one term by the term before it, so 192/256 or 144/192, they would both give you the answer of 0.75. To find n7 (t7 is they way we named it) you use the formula tn=t1xr^(n-1)