Question

Write a recursive sequence that represents the sequence defined by the following explicit formula: an = 2(-2x^2)^n

Answer 1

1) Given this Explicit formula, let's find a_1, and then a_2 and then compare both terms:

`\begin{gathered} a_n=2(-2x^2)^n \\ a_1=2(-2x^2)^1\Rightarrow a_1=-4x^2 \\ a_2=2(-2x^2)^2\Rightarrow a_2=2(4x^4)=8x^4 \\ a_3=2(-2x^2)^3\Rightarrow a_3=2(-8x^6)=-16x^6 \end{gathered}`

2) Comparing both terms we can state that this is a Geometric Sequence we can write its ratio as:

`q=(8x^4)/(-4x^2)=-2x^2`

3) So we can write our Recursive formula as:

`\begin{gathered} a_n=-2x^2* a_(n-1) \\ \text{Testing:} \\ a_2=-2x^2* a_1 \\ a_2=-2x^2*-4x^2 \\ a_{2=\text{ }}8x^4 \end{gathered}`

So the answers are:

`\begin{gathered} a_1=-4x^2 \\ a_n=-2x^2* a_(n-1) \end{gathered}`