1. [2 pt] Prove that for any nN, 1 + 2 + ... + n =...

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1. [2 pt] Prove that for any nN, 1 + 2 + ... + n = (n(n+1). 2. [2 pt] Prove that for any n = N, 61"+9"-1 is divisible by 5. 3. [2 pt] The Tower of Hanoi is a puzzle consisting of three rods and a number of disks of distinct diameters, which can slide onto any rod. The puzzle begins with the disks stacked on one rod in order of decreasing size from the bottom to the top. The objective of the puzzle is to move the entire stack to the last rod, obeying the following rules: Only one disk may be moved at a time. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod. No disk may be placed on top of a disk that is smaller than it. (A) Start (B) Middle (C) Goal Figure 1: Hanoi Tower Puzzle What is the minimal number of moves required to solve a Tower of Hanoi puzzle with n disks? Prove your statement. 1. [2 pt] Prove that for any nN, 1 + 2 + ... + n = (n(n+1). 2. [2 pt] Prove that for any n = N, 61"+9"-1 is divisible by 5. 3. [2 pt] The Tower of Hanoi is a puzzle consisting of three rods and a number of disks of distinct diameters, which can slide onto any rod. The puzzle begins with the disks stacked on one rod in order of decreasing size from the bottom to the top. The objective of the puzzle is to move the entire stack to the last rod, obeying the following rules: Only one disk may be moved at a time. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod. No disk may be placed on top of a disk that is smaller than it. (A) Start (B) Middle (C) Goal Figure 1: Hanoi Tower Puzzle What is the minimal number of moves required to solve a Tower of Hanoi puzzle with n disks? Prove your statement.