Question

Discreet mathematics

Question 19 6 Assemble an inductive proof to show that the quantity 5-1 is a multiple of 4 for all positive integers n Your proof must not contain redundant elements Write your proof on scratch paper first Base Case: Select Select The n-o case is true, since 5' 1 is a multiple F4 The n=1 case true, since 5 1 = 0 is a multiple of 4 The n-2 case true, since 5 -1- 4 is a multiple of 4 Select Finishing the Inductive Step: Select Select This completes the inductive step

Answer 1

• Below is the detailed explanation of the above problem based on induction.
• Answer:

Base case(option B): The n=1 case true, since 5⁰-1 = 0 is a multiple of 4

Inductive Hypothesis(option B): Now assume that we have proved that 5^n-1-1 is a multiple of 4 for some arbitary integer n>=1

Finishing the inductive step(option B): We now substitute n+1 and get that 5ⁿ-1 is a multiple of 4, since 5 mod 4=1.This concludes the proof.

(Option C): (Nothing more needs to be added)

• Explanation:

Proof by induction consists of three step process, first we prove the base case , then we assume for some integer n our statement is correct i.e, inductive hypothesis then at last we prove that for n+1 also our assumption is correct.

To Prove: 5^n-1-1 is divisible by 4

Base case: For n=1 , the statement is true i.e, 5^1-1-1=1-1=0 is divisible by 4.

Inductive hypothesis: For some arbitary integer n>=1 we assume that our statement is correct i.e, 5^n-1-1 is divisible by 4 for some arbitary n>=1.

Finishing inductive step(proof for n+1):  For n+1, we get statement as 5^n+1-1-1=5ⁿ-1, here 5ⁿ-1 is always divisible by 4 as (5ⁿ)%4=1 and subtracting 1 from it will give us 0 so  (5ⁿ-1)%4=0. Hence proved.

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