To solve for [tex]\(a_5\)[/tex] given the initial condition [tex]\(a_1 = 3\)[/tex] and the recurrence relation [tex]\(a_n = -2 a_{n-1}\)[/tex], we can proceed step-by-step by calculating the first few terms in the sequence based on the recurrence relation.
1. We start with the initial term:
[tex]\[
a_1 = 3
\][/tex]
2. Using the recurrence relation [tex]\(a_n = -2 a_{n-1}\)[/tex], we find the second term:
[tex]\[
a_2 = -2 \cdot a_1 = -2 \cdot 3 = -6
\][/tex]
3. Next, we use the recurrence relation to find the third term:
[tex]\[
a_3 = -2 \cdot a_2 = -2 \cdot (-6) = 12
\][/tex]
4. Now, we continue to apply the recurrence relation to find the fourth term:
[tex]\[
a_4 = -2 \cdot a_3 = -2 \cdot 12 = -24
\][/tex]
5. Finally, we use the recurrence relation one more time to find the fifth term:
[tex]\[
a_5 = -2 \cdot a_4 = -2 \cdot (-24) = 48
\][/tex]
Thus, the value of [tex]\(a_5\)[/tex] is [tex]\(48\)[/tex].
