Question

The following function defines a recursive sequence:

f(0) = -2

f(1) = 8

f(n) = -4•f(n -1) - 3•f(n - 2); for n > 1

Which of the following sequences is defined by this recursive function?

-2, 8, -26, -80, …
-2, 8, -26, 80, …
-2, 12, -44, 180, …
-2, -12, -44, -180, …

Answer 1

tried to find the third term in the sequence
by replacing n with 3

Answer 2

\[f(n)=-4 \cdot f(n-1)-3 \cdot f(n-2) \\ \]
I see three n's to replace here with 3
\[f(3)=-4 \cdot f(3-1)-3 \cdot f(3-2) \\ \]
now finish the operations inside the ( )

Answer 3

\[f(3)=-4 \cdot f(2)-3 \cdot f(1)\]
as you see we need f(2)
and f(2) is actually the 3rd term since f(0) is the first term

Answer 4

so let's find the the third term f(2) then we will go back to finding the fourth term f(3)
so replace all the n's with 2's

Answer 5

\[f(n)=-4 \cdot f(n-1)-3 \cdot f(n-2) \\ \]
go ahead
try
replace the n's you see there with 2's
and do nothing else for now

Answer 6

I don't know where to start

Answer 7

i'm telling you exactly where to start

Answer 8

you should see 3 n's
replace the n's there with 2's

Answer 9

well there was one more n

Answer 10

f(2) = -4 * f(2-1) -3 * f(n-2) ?

Answer 11

well there is that last n there you still haven't replaced with 2
\[f(2)=-4 \cdot f(2-1)-3 \cdot f(2-2)\]

Answer 12

now do the operations in the f( )

Answer 13

2-1=?
2-2?

Answer 14

2-1 = 1

2-2 = 0

Answer 15

right so we have
\[f(2)=-4 \cdot f(1)-3 \cdot f(0)\]

Answer 16

but guess what

Answer 17

f(1)
and
f(0)
was given

Answer 18

remember this:
f(0) = -2

f(1) = 8

Answer 19

replace f(1) with 8
replace f(0) with -2

Answer 20

\[f(2)=-4 \cdot f(1)-3 \cdot f(0) \\ f(2)=-4 \cdot 8-3 \cdot -2\]
when folloing the order of operations we know we are to take care of the multiplication then the subtraction
so can you simplify this

Answer 21

-4 * 8 - 3 * -2?

-26

Answer 22

-32+6=-26
sounds good

Answer 23

so far we have the sequence is f(0),f(1),f(2),f(3),f(4),....
where we have -2,8,-26,........

Answer 24

there are two choices you have that fit this so far

Answer 25

so we need to make another round

Answer 26

now let's go back to our f(3)

Answer 27

we already did some simplification
but let's start from the top
to find the 4th term (or the 3-1th term)
replace the n's with 3
\[f(n)=-4 \cdot f(n-1)-3 \cdot f(n-2) \\ \]
whereever you n put a 3
\[f(3)=-4 \cdot f(3-1)-3 \cdot f(3-2) \\ \]

Answer 28

see of you can simplify that
you have to use previous terms in your sequence to finish simplifying

Answer 29

see if you can*

Answer 30

what about the f's?

Answer 31

3-1 is 2

3-2 is 1

Answer 32

you remember before we used what were given about the f's
we were given f(0) and f(1)...
here you have f(2) and f(1)
we just found f(2) and we were given f(1)

Answer 33

right so you have
\[f(3)=-4 \cdot f(2)-3 \cdot f(1)\]
we just found f(2) to be -26
and we were given f(1) to be 8

Answer 34

so is that f(3) = -4 * -26 - 3 * 8

Answer 35

beautiful

Answer 36

80. Answer is B

Answer 37

cool stuff :)

Answer 38

you did well
it just takes a bit of experience with these to get used to them

Answer 39

thank you :)