Solution: Let the score at level $ n $ be $ S_n = S_1 \cdot 2^{n-1} $. Given $ S_3 = S_1 \cdot 4 = 128 \Rightarrow S_1 = 32 $. Then $ S_5 = 32 \cdot 2^{4} = 32 \cdot 16 = 512 $. oxed{512}

["Mastering Geometric Sequences: How to Calculate $ S_n $ and Solve $ S_5 = 512 $ with Ease", "In mathematical problem-solving, especially in sequences, understanding patterns and applying formulas efficiently can save significant time and reduce errors. One common challenge involves geometric sequences—especially when tasked with finding specific terms like $ S_5 $ based on an initial value and a known term such as $ S_3 $.", "### What is $ S_n $ Given in This Problem?", "The problem defines the score at level $ n $ in a geometric sequence by the formula:\n$$\nS_n = S_1 \cdot 2^{n-1}\n$$\nThis shows that $ S_n $ grows exponentially, with $ S_1 $ as the first term and 2 as the common ratio. The pattern doubles at each level.", "### Step 1: Determine $ S_1 $ Using $ S_3 = 128 $", "We’re told that:\n$$\nS_3 = S_1 \cdot 2^{3-1} = S_1 \cdot 2^2 = 4S_1\n$$\nGiven $ S_3 = 128 $, solve for $ S_1 $:\n$$\n4S_1 = 128 \Rightarrow S_1 = \frac{128}{4} = 32\n$$", "### Step 2: Calculate $ S_5 $", "Now that we know $ S_1 = 32 $, use the formula again:\n$$\nS_5 = S_1 \cdot 2^{5-1} = 32 \cdot 2^4 = 32 \cdot 16 = 512\n$$", "### Final Result\n$$\n\boxed{S_5 = 512}\n$$", "### Conclusion: Streamlining Exponentiation in Geometric Sequences", "This structured approach—identifying the pattern, applying the formula, and computing step-by-step—makes solving terms in a geometric sequence intuitive. Memorizing the general form $ S_n = S_1 \cdot 2^{n-1} $ and practicing exponent rules enables quick calculations. Whether studying sequences in algebra, programming algorithms, or financial modeling, mastering such formulas empowers faster and more confident problem solving.", "Key Takeaway:\nUse $ S_n = S_1 \cdot r^{n-1} $ with $ r = 2 $ here, substitute known values, and compute powers stepwise to efficiently reach $ S_5 = 512 $."]