Question

A geometric sequence is shown below.

2, – 6, 18, – 54, 162, ...

Part A:
Write a recursive relationship for this sequence. Explain how you determined your answer.

Part B:
Write an explicit formula for this sequence.

Answer 1

`a_1=2;\ a_2=-6;\ a_3=18;\ a_4=-54;\ a_5=162;\ ...`

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A recursive rule for a geometric sequence:

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

`r=(a_(n+1))/(a_n)\to r=(a_2)/(a_1)=(a_3)/(a_2)=(a_4)/(a_3)=...\\\\r=(-6)/(2)=-3`

Therefore
`\boxed{a_1=2;\qquad a_n=-3a_(n-1)}`

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The exciplit rule:

`a_n=a_1r^(n-1)`

Substitute:

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