This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Observe the recursive construction from the diagrams. Each higher-dimensional figure is built by taking two copies of the previous figure and connecting corresponding dots with edges. The number of vertices V_n in the n-dimensional figure satisfies: V_0 = 1 V_n = 2 · V_n-1 for n ≥ 1 Step 2: Verify with given diagrams. For n=1: V_1 = 2 · V_0 = 2 · 1 = 2 For n=2: V_2 = 2 · V_1 = 2 · 2 = 4 For n=3: V_3 = 2 · V_2 = 2 · 4 = 8 For n=4: V_4 = 2 · V_3 = 2 · 8 = 16 Step 3: Solve the recurrence relation. Unrolling the recursion: V_n = 2 · V_n-1 = 2^2 · V_n-2 = ·s = 2^n · V_0 = 2^n · 1 = 2^n Step 4: This matches the powers of 2 and the number of binary sequences of length n (each dot labels a sequence of 0s and 1s). For the 5-dimensional figure, the number of vertices is V_5 = 2^5. 2^5 = 32 32