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

##### 2 Answers

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

#### Explanation:

A geometric sequence is a sequence with initial term

where the

Dividing any term after the first by the term prior produces

Thus, in the given sequence, the common ratio is

As the initial term is

where

So

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

#### 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

#-16384# ,#4096# ,#-1024# , ...,#1#

Reverse the order:

#1# , ... ,#-1024# ,#4096# ,#-16384#

Express in terms of powers of

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

So there are

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

That is:

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