Question

Which choice is the explicit formula for the following geometric sequence?

0.5, –0.1, 0.02, –0.004, 0.0008, ...

Answer 1

geometric sequence
an=a1(r)^(n-1)
r=common ratio
a1=first term
an=nth term

the common ratio is found by dividing a term by the term before it

-0.1/0.5=-1/5

the common ratio is -1/5
the 1th term is 0.05

the formula is
an=0.5(-1/5)^(n-1) or
an=-2.5(-1/5)^n

Answer 2

`-0.1=-\frac15*0.5`

`0.02=-\frac15*(-0.1)`

`-0.004=-\frac15*0.02`

`0.0008=-\frac15*(-0.004)`

This means the common ratio between the terms in the sequence is
`-\frac15`, so you know the recursive formula is


`a_n=-\frac15a_(n-1)`

which you can solve recursively to find an explicit formula in terms of
`a_1=0.5`.


`a_n=-\frac15a_(n-1)=\left(-\frac15\right)^2a_(n-2)=\left(-\frac15\right)^3a_(n-3)=\cdots=\left(-\frac15\right)^(n-1)a_1`

So the explicit formula is


`a_n=\frac12\left(-\frac15\right)^(n-1)=0.5*(-0.2)^(n-1)`