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Prove each statement using either weak, strong, or structural induction. Make sure to clearly indicate the different parts of your proof: the basis step, the inductive hypothesis, what you will show in the inductive step, and the inductive step. Make sure to clearly format your proofs and to write in complete, clear sentences. EXAMPLE: Prove that for any nonnegative integer n, Σ i = (n+1) Answer: Proof. (by weak induction) Basis step: n = 1 Σ=1 1(1+1)==1 Therefore, (n+1) when n = 1. = Inductive hypothesis: Assume that Inductive step: We will show that i=1 i=1 i= = (+1) for some integer k > 1. i= (k+1)((k+1)+1) k+1 Σ=Σ+ (κ + 1) i=1 By inductive hypothesis, k+1 Σ IME i=1 k(k+1) = +k+1 2 k(k+1)+2(k+1) = 2 (k+2)(k+1) = 2 (k+1)((k+1)+1) 2 Therefore, by weak induction, we have shown that = (n+1) for all nonnegative integers n. Prove that 9" 2" is divisible by 7 for all positive integers n.

Prove each statement using either weak, strong, or structural induction. Make sure to clearly indicate the different parts of your proof: the basis step, the inductive hypothesis, what you will show in the inductive step, and the inductive step. Make sure to clearly format your proofs and to write in complete, clear sentences. EXAMPLE: Prove that for any nonnegative integer n, Σ i = (n+1) Answer: Proof. (by weak induction) Basis step: n = 1 Σ=1 1(1+1)==1 Therefore, (n+1) when n = 1. = Inductive hypothesis: Assume that Inductive step: We will show that i=1 i=1 i= = (+1) for some integer k > 1. i= (k+1)((k+1)+1) k+1 Σ=Σ+ (κ + 1) i=1 By inductive hypothesis, k+1 Σ IME i=1 k(k+1) = +k+1 2 k(k+1)+2(k+1) = 2 (k+2)(k+1) = 2 (k+1)((k+1)+1) 2 Therefore, by weak induction, we have shown that = (n+1) for all nonnegative integers n. Prove that 9" 2" is divisible by 7 for all positive integers n.

Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter21: Simulation
Section21.5: Simulations With Continuous Random Variables
Problem 6P
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Prove each statement using either weak, strong, or structural induction. Make sure to clearly indicate the different parts of your proof: the basis step, the inductive hypothesis, what you will show in the inductive step, and the inductive step. Make sure to clearly format your proofs and to write in complete, clear sentences. EXAMPLE: Prove that for any nonnegative integer n, Σ i = (n+1) Answer: Proof. (by weak induction) Basis step: n = 1 Σ=1 1(1+1)==1 Therefore, (n+1) when n = 1. = Inductive hypothesis: Assume that Inductive step: We will show that i=1 i=1 i= = (+1) for some integer k > 1. i= (k+1)((k+1)+1) k+1 Σ=Σ+ (κ + 1) i=1 By inductive hypothesis, k+1 Σ IME i=1 k(k+1) = +k+1 2 k(k+1)+2(k+1) = 2 (k+2)(k+1) = 2 (k+1)((k+1)+1) 2 Therefore, by weak induction, we have shown that = (n+1) for all nonnegative integers n.
Transcribed Image Text:Prove each statement using either weak, strong, or structural induction. Make sure to clearly indicate the different parts of your proof: the basis step, the inductive hypothesis, what you will show in the inductive step, and the inductive step. Make sure to clearly format your proofs and to write in complete, clear sentences. EXAMPLE: Prove that for any nonnegative integer n, Σ i = (n+1) Answer: Proof. (by weak induction) Basis step: n = 1 Σ=1 1(1+1)==1 Therefore, (n+1) when n = 1. = Inductive hypothesis: Assume that Inductive step: We will show that i=1 i=1 i= = (+1) for some integer k > 1. i= (k+1)((k+1)+1) k+1 Σ=Σ+ (κ + 1) i=1 By inductive hypothesis, k+1 Σ IME i=1 k(k+1) = +k+1 2 k(k+1)+2(k+1) = 2 (k+2)(k+1) = 2 (k+1)((k+1)+1) 2 Therefore, by weak induction, we have shown that = (n+1) for all nonnegative integers n.
Prove that 9" 2" is divisible by 7 for all positive integers n.
Transcribed Image Text:Prove that 9" 2" is divisible by 7 for all positive integers n.
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