Sure, let's solve for [tex]\( f(1) \)[/tex] using the provided sequence information.
We are given the sequence formula:
[tex]\[ f(n+1) = f(n) - 3 \][/tex]
And the specific value:
[tex]\[ f(4) = 22 \][/tex]
We need to find [tex]\( f(1) \)[/tex]. We'll work backward from [tex]\( f(4) \)[/tex].
First, let's find [tex]\( f(3) \)[/tex]:
[tex]\[ f(4) = f(3) - 3 \][/tex]
Substitute [tex]\( f(4) = 22 \)[/tex]:
[tex]\[ 22 = f(3) - 3 \][/tex]
Add 3 to both sides to solve for [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = 22 + 3 = 25 \][/tex]
Next, let's find [tex]\( f(2) \)[/tex]:
[tex]\[ f(3) = f(2) - 3 \][/tex]
Substitute [tex]\( f(3) = 25 \)[/tex]:
[tex]\[ 25 = f(2) - 3 \][/tex]
Add 3 to both sides to solve for [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 25 + 3 = 28 \][/tex]
Finally, let's find [tex]\( f(1) \)[/tex]:
[tex]\[ f(2) = f(1) - 3 \][/tex]
Substitute [tex]\( f(2) = 28 \)[/tex]:
[tex]\[ 28 = f(1) - 3 \][/tex]
Add 3 to both sides to solve for [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 28 + 3 = 31 \][/tex]
Thus, the value of [tex]\( f(1) \)[/tex] is:
[tex]\[ \boxed{31} \][/tex]
