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

1 Answer
Alan P.
Jan 13, 2017

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

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