2. Let P (n) be the statement that21 1 1 11 ... 24 9 n n , where n is an integer greaterthan 1.a) What is the statement P (2)?b) Show that P (2) is true, completing the basis step of the proof.c) What is the inductive hypothesis?d) What do you need to prove in the inductive step?e) Complete the inductive step.
Question
- Let P (n) be the statement that21 1 1 11 ... 24 9 n n , where n is an integer greaterthan 1.a) What is the statement P (2)?b) Show that P (2) is true, completing the basis step of the proof.c) What is the inductive hypothesis?d) What do you need to prove in the inductive step?e) Complete the inductive step.
Solution
a) The statement P(2) is 1/2 + 1/3 < 2 - 1/2.
b) To show that P(2) is true, we need to prove that 1/2 + 1/3 < 2 - 1/2. Simplifying the right side of the inequality gives us 1.5. Simplifying the left side of the inequality gives us 5/6, which is approximately 0.833. Since 0.833 is less than 1.5, P(2) is true.
c) The inductive hypothesis is the assumption that P(k) is true for some arbitrary positive integer k. That is, we assume that 1/2 + 1/3 + ... + 1/k < k - 1/k.
d) In the inductive step, we need to prove that if P(k) is true, then P(k+1) is also true. That is, we need to show that if 1/2 + 1/3 + ... + 1/k < k - 1/k, then 1/2 + 1/3 + ... + 1/k + 1/(k+1) < (k+1) - 1/(k+1).
e) To complete the inductive step, we start with the inductive hypothesis and add 1/(k+1) to both sides: 1/2 + 1/3 + ... + 1/k + 1/(k+1) < k - 1/k + 1/(k+1). We need to show that the right side is less than (k+1) - 1/(k+1). Simplifying the right side gives us k + 1/k - 1/(k+1). Since k > 1, we know that 1/k is greater than 1/(k+1), so k + 1/k - 1/(k+1) is less than k + 1/k, which is less than (k+1) - 1/(k+1). Therefore, if P(k) is true, then P(k+1) is also true, completing the inductive step.
Similar Questions
1. Let P (n) be the statement that 2 2 2 2 1 2 11 2 3 ... 6n n nn for the positiveinteger n.a) What is the statement P (1)?b) Show that P (1) is true, completing the basis step of the proof.c) What is the inductive hypothesis?d) What do you need to prove in the inductive step?e) Complete the inductive step, identifying where you use the inductive hypothesis.
17) Why is the inductive hypothesis important in mathematical induction?a) It calculates numerical valuesb) It sovles for unknown variablesc) It provides a basis for proving the inductive stepd) It disproves the given statement
16) What is proven in the inductive step of mathematical induction?a) The statement holds for all real numbers.b) The statement holds for an arbitrary case.c) The statement holds for the next case, given it holds for a particular case.d) The statement holds for the base case.
15) In mathematical induction, what is the purpose of the base case?a) To provide a general formulab) To disprove the given statementc) To provide a starting point for the inductiond) To solve for unknown variables
Assume that N = {0, 1, 2, 3, . . .}.(a) Suppose that we change the induction mechanism as follows:• Base case: Prove that P (0) is true• Inductive step: Prove that for all k ≥ 0, P (k) ⇒ P (k + 2)Explain why this would not constitute a valid proof that P (n) is true for alln ∈ N. How would you change the base case to obtain a valid proof?
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