Let's solve this problem step by step.
1. Understand the initial condition and recurrence relation:
- We are given [tex]\( f(1) = 100 \)[/tex].
- The sequence is defined recursively by the formula [tex]\( f(n+1) = f(n) - 8 \)[/tex].
2. Calculate the next terms in the sequence manually:
- Start from the initial term [tex]\( f(1) = 100 \)[/tex].
- Compute the subsequent terms using the recurrence relation.
- Calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = f(1) - 8 = 100 - 8 = 92
\][/tex]
- Calculate [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = f(2) - 8 = 92 - 8 = 84
\][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[
f(4) = f(3) - 8 = 84 - 8 = 76
\][/tex]
- Calculate [tex]\( f(5) \)[/tex]:
[tex]\[
f(5) = f(4) - 8 = 76 - 8 = 68
\][/tex]
- Calculate [tex]\( f(6) \)[/tex]:
[tex]\[
f(6) = f(5) - 8 = 68 - 8 = 60
\][/tex]
3. Verify the sequence consistency:
- Each step follows the recurrence relation [tex]\( f(n+1) = f(n) - 8 \)[/tex] correctly and leads to consistent results.
Thus, the value of [tex]\( f(6) \)[/tex] is [tex]\(\boxed{60}\)[/tex].
