Question

A recursive rule for a geometric sequence is a1=3;an=1/2an−1.

What is the explicit rule for this sequence?



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Answer 1

The answer is: a_n = 6*(1/2)^n

Answer 2

A recursive rule for a geometric sequence:

`a_1\\\\a_n=r\cdot a_(n-1)`

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`a_1=3\\\\a_n=(1)/(2)a_(n-1)\to \boxed{r=(1)/(2)}`

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Exciplit rule:

`a_n=a_1r^(n-1)`

Substitute:

`a_n=3\left((1)/(2)\right)^(n-1)=3\cdot\left((1)/(2)\right)^n\cdot\left((1)/(2)\right)^(-1)=3\cdot\left((1)/(2)\right)^n\cdot2\\\\\boxed{a_n=6\cdot\left((1)/(2)\right)^n}`