How do you write an equation for the nth term of the geometric sequence #4, 8, 16, ...#?

1 Answer
Jan 13, 2017

Answer:

#a_n=4 * 2^(n-1)#
...but see below

Explanation:

It seems that currently the most common usage is that the initial value of a sequence is considered the "first" value.
That is for the given geometric sequence: 4, 8, 16, ...
#color(white)("XXX")a_1=4#
Each term after the first increases the immediately prior term by a factor of #2# and since there are #(n-1)# terms after the first for the #n^(th)# term, we have
#color(white)("XXX")a_n = a_1 xx underbrace(2 xx 2 xx ... xx2)_(n" times") = a_1 * 2^(n-1) = 4 * 2^(n-1)#

Less common these days (but the version I learned long ago) was to call the initial value of the sequence, the #"zero"^(th)# value.
That is for the given sequence: 4, 8, 16, ...
#color(white)("XXX")a_0=4#
and the equation for the #n^(th)# value would be
#color(white)("XXX")a_n=a_0 * 2^n# or #4 * 2^n#