Question: Create a math video explanation for a beginner student. We are given two sequences: $W_0 = 1$ $W_{n+1} = 2W_n + 4$ and $S_n = -3W_n + 4$ Explain specifically how to find the formula for $S_{n+1}$. Show every step visually and explain why we replace $n$ with $n+1$. Start with: $S_n = -3W_n + 4$ Then show: $S_{n+1} = -3W_{n+1} + 4$ Explain that $W_{n+1}$ is known from the first recurrence: $W_{n+1} = 2W_n + 4$ Substitute this into $S_{n+1}$: $S_{n+1} = -3(2W_n + 4) + 4$ Distribute $-3$: $S_{n+1} = -6W_n - 12 + 4$ Simplify: $S_{n+1} = -6W_n - 8$ Make the animation beginner-friendly. Clearly highlight the part that changes from $S_n$ to $S_{n+1}$, and explain why $W_{n+1}$ is replaced by $2W_n + 4$.

1. **State the problem:** We have two sequences defined as: $$W_0 = 1$$ $$W_{n+1} = 2W_n + 4$$ and $$S_n = -3W_n + 4$$ We want to find a formula for $$S_{n+1}$$. 2. **Understand the notation:** The subscript $$n$$ represents the position in the sequence. To find $$S_{n+1}$$, we replace every $$n$$ in the formula for $$S_n$$ with $$n+1$$. 3. **Write the formula for $$S_{n+1}$$:** $$S_{n+1} = -3W_{n+1} + 4$$ 4. **Use the recurrence for $$W_{n+1}$$:** From the first sequence, we know: $$W_{n+1} = 2W_n + 4$$ 5. **Substitute $$W_{n+1}$$ into $$S_{n+1}$$:** $$S_{n+1} = -3(2W_n + 4) + 4$$ 6. **Distribute the $$-3$$:** $$S_{n+1} = -6W_n - 12 + 4$$ 7. **Simplify the constants:** $$S_{n+1} = -6W_n - 8$$ **Explanation:** - We replace $$n$$ with $$n+1$$ to find the next term in the sequence. - The term $$W_{n+1}$$ is replaced using its recurrence relation to express it in terms of $$W_n$$. - This substitution helps us write $$S_{n+1}$$ in terms of $$W_n$$, which is useful for further analysis or computation. This step-by-step process clearly shows how to find $$S_{n+1}$$ from the given sequences.