Question

How do you find the number of terms in the following geometric sequence: -409.6, 102.4, -25.6,..., 0.025?

Answer 1

Solve for the common ratio, and figure out how many times the initial term is multiplied by it to reach the final term. Doing so shows that there are `8` terms.

Explanation:

A geometric sequence is a sequence with initial term `a` and common ratio `r` of the form
`a, ar, ar^2, ar^3, ..., ar^n, ...`
where the `n^(th)` term is `ar^(n-1)`.

Dividing any term after the first by the term prior produces `r`, as

`(ar^(n))/(ar^(n-1)) = r`

Thus, in the given sequence, the common ratio is

`r = 102.4/(-409.6) = -0.25 = -1/4`

As the initial term is `-409.6` and the final term is `0.025` we have

`0.025 = -409.6(-0.25)^(n-1)`

`=> (-0.25)^(n-1) = 0.025/(-409.6) ~~-0.000061`

where `n` is the number of terms in the sequence. We could solve for `n` algebraically using logarithms, or simply by taking successive powers of `-0.25` and seeing how many we need.

`(-0.25)^2 = 0.0625`

`(-0.25)^3 = -0.015625`

`(-0.25)^4 = 0.00390625`

`(-0.25)^5 = -0.0009765625`

`(-0.25)^6 = 0.000244140625`

`(-0.25)^7 ~~-0.000061`

So `n-1 = 7` meaning the sequence has `8` terms.

Answer 2

Transform the sequence into one where the count of terms is easier to spot, viz `8`.

Explanation:

Let's perform some transformations on the sequence that will keep it a geometric sequence with the same number of terms, but make the answer easier to spot:

Start with:

`-409.6`, `102.4`, `-25.6`, ... , `0.025`

Multiply by `40`:

`-16384`, `4096`, `-1024`, ..., `1`

Reverse the order:

`1`, ... , `-1024`, `4096`, `-16384`

Express in terms of powers of `4`:

`4^0`, ... , `-4^5`, `4^6`, `-4^7`

So there are `8` terms:

`4^0`, `-4^1`, `4^2`, `-4^3`, `4^4`, `-4^5`, `4^6`, `-4^7`

That is:

`(-4)^0`, `(-4)^1`, ... , `(-4)^7`