What is the third step in proving by mathematical induction that for every positive integer $n , 11^n-6$ is divisible by 5, is true?
A. Assume that $11^{k+1}-6$ is divisible by 5.
B. Show that $11^{k+1}-6$ is divisible by 5.
C. Assume that $11^1-6$ is divisible by 5.
D. Show that $11^k-6$ is divisible by 5.
Answer (2)
The problem requires identifying the third step in a proof by mathematical induction.
The first step is the base case.
The second step is the inductive hypothesis.
The third step is the inductive step, which involves showing that if the statement is true for n = k , then it is also true for n = k + 1 . The answer is to show that 1 1 k + 1 − 6 is divisible by 5.
Show that 1 1 k + 1 − 6 is divisible by 5 .
Explanation
Understanding Mathematical Induction The problem asks for the third step in a proof by mathematical induction that 1 1 n − 6 is divisible by 5 for every positive integer n . Mathematical induction involves three main steps: 1) Base case: verifying the statement for an initial value (usually n = 1 ), 2) Inductive hypothesis: assuming the statement holds for some integer k , and 3) Inductive step: proving the statement holds for k + 1 based on the assumption for k .
Base Case The first step is the base case, which involves showing that the statement is true for n = 1 . In this case, we would show that 1 1 1 − 6 is divisible by 5.
Inductive Hypothesis The second step is the inductive hypothesis, where we assume that the statement is true for some positive integer k . That is, we assume that 1 1 k − 6 is divisible by 5.
Inductive Step The third step is the inductive step, where we must show that if the statement is true for n = k , then it is also true for n = k + 1 . In other words, we need to show that 1 1 k + 1 − 6 is divisible by 5, assuming that 1 1 k − 6 is divisible by 5.
Identifying the Third Step Therefore, the third step in the proof by mathematical induction is to show that 1 1 k + 1 − 6 is divisible by 5.
Final Answer The correct answer is 'Show that 1 1 k + 1 − 6 is divisible by 5'.
Examples
Mathematical induction is a powerful technique used to prove statements about all positive integers. For example, you can use mathematical induction to prove that the sum of the first n positive integers is 2 n ( n + 1 ) for all positive integers n . This technique is widely used in computer science to prove the correctness of algorithms and data structures.
In the process of proving by mathematical induction that 1 1 n − 6 is divisible by 5 for all positive integers n, the third step is to demonstrate that 1 1 k + 1 − 6 is divisible by 5 under the assumption that 1 1 k − 6 is divisible by 5. Thus, the correct answer is to show this divisibility. The answer choice is: B. Show that 1 1 k + 1 − 6 is divisible by 5.
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