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].
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].
