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

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.

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#