1. **Problem statement:** Show that the sequences $\{u_n\}$ satisfy the recurrence relation $$u_n = -3u_{n-1} + 4u_{n-2}$$ for each given sequence. 2. **Recurrence relation formula:** $$u_n = -3u_{n-1} + 4u_{n-2}$$ means each term depends on the two previous terms. 3. **Check each sequence:** **(a) $u_n = 0$** - Substitute: $u_n = 0$, $u_{n-1} = 0$, $u_{n-2} = 0$ - Check: $0 \stackrel{?}{=} -3(0) + 4(0) = 0$ - True, so sequence satisfies the relation. **(b) $u_n = 1$** - Substitute: $u_n = 1$, $u_{n-1} = 1$, $u_{n-2} = 1$ - Check: $1 \stackrel{?}{=} -3(1) + 4(1) = -3 + 4 = 1$ - True, so sequence satisfies the relation. **(c) $u_n = (-4)^n$** - Substitute: $u_n = (-4)^n$, $u_{n-1} = (-4)^{n-1}$, $u_{n-2} = (-4)^{n-2}$ - Check: $$(-4)^n \stackrel{?}{=} -3(-4)^{n-1} + 4(-4)^{n-2}$$ - Factor $(-4)^{n-2}$: $$(-4)^n = (-4)^{n-2} \times (-4)^2 = (-4)^{n-2} \times 16$$ - Right side: $$-3(-4)^{n-1} + 4(-4)^{n-2} = (-4)^{n-2}(-3(-4) + 4) = (-4)^{n-2}(12 + 4) = 16(-4)^{n-2}$$ - Both sides equal $16(-4)^{n-2}$, so true. **(d) $u_n = 2(-4)^n + 3$** - Substitute: $$u_n = 2(-4)^n + 3$$ $$u_{n-1} = 2(-4)^{n-1} + 3$$ $$u_{n-2} = 2(-4)^{n-2} + 3$$ - Check: $$2(-4)^n + 3 \stackrel{?}{=} -3[2(-4)^{n-1} + 3] + 4[2(-4)^{n-2} + 3]$$ - Expand right side: $$= -6(-4)^{n-1} - 9 + 8(-4)^{n-2} + 12 = -6(-4)^{n-1} + 8(-4)^{n-2} + 3$$ - Factor $(-4)^{n-2}$: $$-6(-4)^{n-1} + 8(-4)^{n-2} = (-4)^{n-2}(-6(-4) + 8) = (-4)^{n-2}(24 + 8) = 32(-4)^{n-2}$$ - So right side is: $$32(-4)^{n-2} + 3$$ - Left side: $$2(-4)^n + 3 = 2(-4)^{n-2}(-4)^2 + 3 = 2(-4)^{n-2} \times 16 + 3 = 32(-4)^{n-2} + 3$$ - Both sides equal, so true. 4. **Determine which sequences are arithmetic or geometric:** - Arithmetic sequence: constant difference $d = u_n - u_{n-1}$ - Geometric sequence: constant ratio $r = \frac{u_n}{u_{n-1}}$ (a) $u_n=0$ is constant, so arithmetic with $d=0$ and geometric with $r$ undefined but can be considered geometric with ratio 1. (b) $u_n=1$ constant, arithmetic with $d=0$, geometric with $r=1$. (c) $u_n=(-4)^n$ geometric with ratio $r = -4$. (d) $u_n=2(-4)^n + 3$ is neither arithmetic nor geometric because of the added constant 3. 5. **Find the sum of the series $S = \sum_{n=0}^\infty u_n$ for each sequence:** - For infinite sums, convergence is required. (a) $u_n=0$: sum is 0. (b) $u_n=1$: sum diverges (infinite). (c) $u_n=(-4)^n$: diverges because $|r|=4>1$. (d) $u_n=2(-4)^n + 3$: diverges because of geometric term with $|r|=4>1$ and constant 3. --- **Final answers:** - All sequences satisfy the recurrence. - Arithmetic: (a), (b) - Geometric: (a), (b), (c) - Series sums: (a) 0, (b) diverges, (c) diverges, (d) diverges