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

Dec 19, 2015

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
$a , a r , a {r}^{2} , a {r}^{3} , \ldots , a {r}^{n} , \ldots$
where the ${n}^{t h}$ term is $a {r}^{n - 1}$.

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

$\frac{a {r}^{n}}{a {r}^{n - 1}} = r$

Thus, in the given sequence, the common ratio is

$r = \frac{102.4}{- 409.6} = - 0.25 = - \frac{1}{4}$

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

$0.025 = - 409.6 {\left(- 0.25\right)}^{n - 1}$

$\implies {\left(- 0.25\right)}^{n - 1} = \frac{0.025}{- 409.6} \approx - 0.000061$

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.

${\left(- 0.25\right)}^{2} = 0.0625$

${\left(- 0.25\right)}^{3} = - 0.015625$

${\left(- 0.25\right)}^{4} = 0.00390625$

${\left(- 0.25\right)}^{5} = - 0.0009765625$

${\left(- 0.25\right)}^{6} = 0.000244140625$

${\left(- 0.25\right)}^{7} \approx - 0.000061$

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

Dec 19, 2015

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:

$- 409.6$, $102.4$, $- 25.6$, ... , $0.025$

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:

${\left(- 4\right)}^{0}$, ${\left(- 4\right)}^{1}$, ... , ${\left(- 4\right)}^{7}$