## anonymous one year ago The following function defines a recursive sequence: f(0) = -3 f(1) = 6 f(n) = -2•f(n -1) - f(n - 2); for n > 1 Which of the following sequences is defined by this recursive function?

1. anonymous

-3, 6, -9, -12, … -3, 20, -95, 480, … -3, 6, -9, 12, … -3, -20, -95, -480, …

2. SolomonZelman

I will change my notations to $$a_{n}$$ if you don't mind, ok?

3. anonymous

um okay i guess

4. SolomonZelman

But you are given the first two terms $$a_0$$ and $$a_1$$ so just based on that you can exclude the rest of the options

5. SolomonZelman

u have only 2 possible options after doing elimination, right? and they are A and C

6. SolomonZelman

$$\large\color{black}{ \displaystyle a_n=-2\cdot \left(a_{n-1}\right) -\left(a_{n-2}\right) }$$ $$\large\color{black}{ \displaystyle a_2=-2\cdot \left(a_{2-1}\right) -\left(a_{2-2}\right) }$$ $$\large\color{black}{ \displaystyle a_2=-2\cdot \left(a_{1}\right) -\left(a_{0}\right) }$$ $$\large\color{black}{ \displaystyle a_2=-2\cdot \left(6\right) -\left(-3\right) }$$ $$\large\color{black}{ \displaystyle a_2=-12 +3 }$$ $$\large\color{black}{ \displaystyle a_2=-9}$$

7. SolomonZelman

then, you can find $$a_3$$ using the same formula

8. anonymous

so its c?

9. SolomonZelman

let me see.... $$\large\color{black}{ \displaystyle a_3=-2\cdot \left(a_{3-1}\right) -\left(a_{3-2}\right) }$$ $$\large\color{black}{ \displaystyle a_2=-2\cdot \left(a_{2}\right) -\left(a_{1}\right)=-2\cdot(-9)-(6)=18-6=12 }$$

10. SolomonZelman

yes C is right

11. SolomonZelman

you can deduce that by logic, that you have a negative term $$a_{n-1}$$ (in this case a negative term $$a_{2}$$ (which is =-9) |--> so when multiplied times -2 it becomes twice as much and positive. This negative term $$a_2$$ has a greater absolute value (or greater magnitude) than $$a_{n-2}$$ (and $$a_{n-2}$$ in this case is $$a_1$$ which is 6) So you are having a case $$a_3=-2\times ({\rm -greater})~-~({\rm smaller})~~~~~~~\Rightarrow~\rm positive$$

12. anonymous

okay great thank you so much for your help! : )

13. SolomonZelman

and you know that it is 12 and -12, having eliminated every option besides A and C. And since the result (for $$a_2$$) must be positive, therefore it is 12 (not -12), and thus the answer is C.

14. SolomonZelman

this is kidnd of an implicit.... yw

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