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Given \( f(0)=2 \) and \( f(1)=4 \), such that \( f(x+2)=4f(x)-f(x+1) \), what is \( f(5) \)?

A. 12
B. 44
C. 45
D. 4

Answer :

To solve the problem, we'll use the given initial conditions and the recurrence relation to find [tex]\( f(5) \)[/tex].

We have:
- [tex]\( f(0) = 2 \)[/tex]
- [tex]\( f(1) = 4 \)[/tex]

The recurrence relation is:
[tex]\[ f(x+2) = 4f(x) - f(x+1) \][/tex]

Let's calculate each step:

1. Calculate [tex]\( f(2) \)[/tex]:

Use the recurrence relation with [tex]\( x = 0 \)[/tex]:
[tex]\[
f(2) = 4f(0) - f(1)
\][/tex]
Substitute the known values:
[tex]\[
f(2) = 4(2) - 4 = 8 - 4 = 4
\][/tex]

2. Calculate [tex]\( f(3) \)[/tex]:

Use the recurrence relation with [tex]\( x = 1 \)[/tex]:
[tex]\[
f(3) = 4f(1) - f(2)
\][/tex]
Substitute the known values:
[tex]\[
f(3) = 4(4) - 4 = 16 - 4 = 12
\][/tex]

3. Calculate [tex]\( f(4) \)[/tex]:

Use the recurrence relation with [tex]\( x = 2 \)[/tex]:
[tex]\[
f(4) = 4f(2) - f(3)
\][/tex]
Substitute the known values:
[tex]\[
f(4) = 4(4) - 12 = 16 - 12 = 4
\][/tex]

4. Calculate [tex]\( f(5) \)[/tex]:

Use the recurrence relation with [tex]\( x = 3 \)[/tex]:
[tex]\[
f(5) = 4f(3) - f(4)
\][/tex]
Substitute the known values:
[tex]\[
f(5) = 4(12) - 4 = 48 - 4 = 44
\][/tex]

So, [tex]\( f(5) \)[/tex] is [tex]\( 44 \)[/tex], which corresponds to option [tex]\( B \)[/tex].

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