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.
Question
- 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.
Solution
a) The statement P(1) is 1^2 = 6*1.
b) To show that P(1) is true, we substitute n=1 into the equation. The left side of the equation becomes 1^2 which equals 1. The right side of the equation becomes 6*1 which also equals 1. Therefore, P(1) is true.
c) The inductive hypothesis is the assumption that P(k) is true for some positive integer k. That is, we assume that 1^2 + 2^2 + 3^2 + ... + k^2 = 6k.
d) In the inductive step, we need to prove that P(k+1) is true, assuming that P(k) is true. That is, we need to show that 1^2 + 2^2 + 3^2 + ... + k^2 + (k+1)^2 = 6(k+1).
e) To complete the inductive step, we start with the left side of P(k+1) which is 1^2 + 2^2 + 3^2 + ... + k^2 + (k+1)^2. We can split this into (1^2 + 2^2 + 3^2 + ... + k^2) + (k+1)^2. By the inductive hypothesis, we know that 1^2 + 2^2 + 3^2 + ... + k^2 = 6k. So, we substitute this into the equation to get 6k + (k+1)^2. Simplifying this gives us 6k + k^2 + 2k + 1. This simplifies further to k^2 + 8k + 1. The right side of P(k+1) is 6(k+1) which simplifies to 6k + 6. Since the left side and the right side of P(k+1) are equal, we have shown that P(k+1) is true assuming that P(k) is true. Therefore, by mathematical induction, the statement P(n) is true for all positive integers n.
Similar Questions
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.
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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