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# The function f(n) = 3f(n-2) - 2f(n-1), where f(2) = 3 and f(1) = -1. What is the value of f(5)?

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The function f(n) = 3f(n-2) - 2f(n-1), where f(2) = 3 and f(1) = -1. What is the value of f(5)?

Feb 6, 2020

#1
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f(n)   =   3f(n - 2)  -  2f(n - 1)        If we plug in  5  for n  we get...

f(5)   =   3f(5 - 2)  -  2f(5 - 1)   =   3f(3)  -  2f(4)

Now lets work on finding  f(3)  and  f(4)

We're given:

f(1)   =   -1

f(2)   =   3

To find  f(3) ,  plug in  3  for  n  into the function:

f(n)   =   3f(n - 2)  -  2f(n - 1)

f(3)   =   3f(3 - 2)  -  2f(3 - 1)

f(3)   =   3f( 1 )  -  2f( 2 )            We know   f(1)  =  -1   and   f(2)  =  3   so we can say...

f(3)   =   3( -1 )  -  2( 3 )

f(3)   =   -9

To find  f(4) ,  plug in  4  for  n  into the function:

f(n)   =   3f(n - 2)  -  2f(n - 1)

f(4)   =   3f(4 - 2)  -  2f(4 - 1)

f(4)   =   3f( 2 )  -  2f( 3 )            We know   f(2)  =  3   and   f(3)  =  -9   so we can say...

f(4)   =   3(  3 )  -  2( -9 )

f(4)   =   27

Now that we know   f(3)  =  -9   and   f(4)  =  27   we can find   f(5):

f(n)   =   3f(n - 2)  -  2f(n - 1)

f(5)   =   3f(5 - 2)  -  2f(5 - 1)

f(5)   =   3f( 3 )  -  2f( 4 )            We know   f(3)  =  -9   and   f(4)  =  27   so we can say...

f(5)   =   3( -9 )  -  2( 27 )

f(5)   =   -81

Feb 6, 2020
#2
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The function f(n) = 3f(n-2) - 2f(n-1), where f(2) = 3 and f(1) = -1.
What is the value of f(5)?

$$\begin{array}{|rcll|} \hline \mathbf{f(n)} &=& \mathbf{(-1)^n*3^{n-1}} \\\\ f(5) &=& (-1)^5*3^{5-1} \\ f(5) &=& -3^4 \\ \mathbf{f(5)} &=& \mathbf{-81} \\ \hline \end{array}$$ Feb 6, 2020