To find [tex]\( f(4) \)[/tex] given the conditions [tex]\( f(1) = 160 \)[/tex] and the recursive relation [tex]\( f(n+1) = -2 f(n) \)[/tex], we will compute each term step-by-step.
1. Start with the initial value:
[tex]\[
f(1) = 160
\][/tex]
2. Compute [tex]\( f(2) \)[/tex] using the recursive relation:
[tex]\[
f(2) = -2 \cdot f(1)
\][/tex]
Substituting [tex]\( f(1) = 160 \)[/tex]:
[tex]\[
f(2) = -2 \cdot 160 = -320
\][/tex]
3. Compute [tex]\( f(3) \)[/tex] using the recursive relation:
[tex]\[
f(3) = -2 \cdot f(2)
\][/tex]
Substituting [tex]\( f(2) = -320 \)[/tex]:
[tex]\[
f(3) = -2 \cdot (-320) = 640
\][/tex]
4. Compute [tex]\( f(4) \)[/tex] using the recursive relation:
[tex]\[
f(4) = -2 \cdot f(3)
\][/tex]
Substituting [tex]\( f(3) = 640 \)[/tex]:
[tex]\[
f(4) = -2 \cdot 640 = -1280
\][/tex]
Therefore, the value of [tex]\( f(4) \)[/tex] is:
[tex]\[
f(4) = -1280
\][/tex]
