Question

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?

Answer 1

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

Answer 2

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

Answer 3

um okay i guess

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

then, you can find \(a_3\) using the same formula

Answer 8

so its c?

Answer 9

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

Answer 10

yes C is right

Answer 11

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
\(a_3=-2\times ({\rm -greater})~-~({\rm smaller})~~~~~~~\Rightarrow~\rm positive\)

Answer 12

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

Answer 13

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.

Answer 14

this is kidnd of an implicit....
yw