High School

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For a sequence defined by [tex]f(1) = 13[/tex] and [tex]f(n) = 2f(n-1) + (n-2)[/tex], which of the following is the value of [tex]f(4)[/tex]?

1. 53
2. 76
3. 108
4. 126

Answer :

We are given the recurrence relation

[tex]$$
f(n)=2 f(n-1)+(n-2)
$$[/tex]

with the initial condition

[tex]$$
f(1)=13.
$$[/tex]

We want to find [tex]\( f(4) \)[/tex].

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

Substitute [tex]\( n = 2 \)[/tex] into the recurrence:

[tex]$$
f(2)=2f(1)+(2-2)=2\cdot 13+0=26.
$$[/tex]

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

Substitute [tex]\( n = 3 \)[/tex]:

[tex]$$
f(3)=2f(2)+(3-2)=2\cdot 26+1=52+1=53.
$$[/tex]

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

Substitute [tex]\( n = 4 \)[/tex]:

[tex]$$
f(4)=2f(3)+(4-2)=2\cdot 53+2=106+2=108.
$$[/tex]

Thus, the value of [tex]\( f(4) \)[/tex] is

[tex]$$
\boxed{108}.
$$[/tex]

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