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

0.5, –0.1, 0.02, –0.004, 0.0008, ...

Respuesta :

LammettHash
[tex]-0.1=-\dfrac15\times0.5[/tex]
[tex]0.02=-\dfrac15\times(-0.1)[/tex]
[tex]-0.004=-\dfrac15\times0.02[/tex]
[tex]0.0008=-\dfrac15\times(-0.004)[/tex]

This means the common ratio between the terms in the sequence is [tex]-\dfrac15[/tex], so you know the recursive formula is

[tex]a_n=-\dfrac15a_{n-1}[/tex]

which you can solve recursively to find an explicit formula in terms of [tex]a_1=0.5[/tex].

[tex]a_n=-\dfrac15a_{n-1}=\left(-\dfrac15\right)^2a_{n-2}=\left(-\dfrac15\right)^3a_{n-3}=\cdots=\left(-\dfrac15\right)^{n-1}a_1[/tex]

So the explicit formula is

[tex]a_n=\dfrac12\left(-\dfrac15\right)^{n-1}=0.5\times(-0.2)^{n-1}[/tex]
apologiabiology
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

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