We are trying to prove that a proposition P(n) is true for all n > 0 using mathematical induction. Suppose we are trying to prove the k + 1 case, P(k+ 1), using strong mathematical induction. What are we allowed to do which we wouldn't do if we had been using weak mathematical induction? We are able to prove P(k+1) as an inequality instead of just an equality. We do not need to make an assumption about the lower bound for k (for example, we do not need to assume that k > 0). We can prove that P(k+1) → P(k+2) instead of P(k) → P(k + 1). We can use the fact that P(k-1) is true, for as long as (k − 1) > 0.

We are trying to prove that a proposition P(n) is true for all n > 0 using mathematical induction. Suppose we are trying to prove the k + 1 case, P(k+ 1), using strong mathematical induction. What are we allowed to do which we wouldn't do if we had been using weak mathematical induction? We are able to prove P(k+1) as an inequality instead of just an equality. We do not need to make an assumption about the lower bound for k (for example, we do not need to assume that k > 0). We can prove that P(k+1) → P(k+2) instead of P(k) → P(k + 1). We can use the fact that P(k-1) is true, for as long as (k − 1) > 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

R2
The remaining problems require you to construct a mathematical proof by induction. Remember, if you don't make use of the inductive hypothesis, there is no way to finish the proof correctly! 4. Prove the following formula for all integers n ≥ 1: 1² +2²+3² +4² +.. .+ n² = n(n+1)(2n+1) 6
Let x be a real number. Given the following statement (M): “x* > x vx > 1". Which among the following options is (are) true? (M) is true and we can prove it by using Strong induction. O (M) is a false statement. (M) is true and we can prove it by O using the first principle of Mathematical induction. O None of these (M) is true and both principles of Mathematical induction can be used to prove it.
Let P(n) be the proposition "(n + 2)(n + 1)n is divisible by both 2 and 3. is true for all integers n > 1.
I really need help with this discrete mathematics/proofs question. Please show all work clearly. Please find attached. Thank you so much!
In each of the following statements, n represents a positive integer. One of the statements is true and the other two are false. (A) If n! is even, then n is not prime. (B) If n is a square number, then n is not prime. (C) If n is a prime number and n ≥ 10, then n + 2 is prime or n +4 is prime. (a) Write down which two statements are false and in each case give a counter-example to show that it is false. (b) Prove the true statement.
All answers should be justified
DISCRETE MATHEMATICSProve that for every integer n where n is greater than or equal to 3, P(n+1, 3) - P(n, 3) = 3P(n, 2). For full credit you must use the factorial definition of a permutation. You must give your proof line-by-line, with each line a statement with its justification. You must show explicit, formal start and termination statements for the overall proof. In English it would be: For all integers n greater than or equal to 3, the 3-permutation of n+1 elements minus the 3-permutation of n elements is equal to three times the 2-permutation of n elements.
please send step by step handwritten proof of Q10
Suppose you know the following about a statement P(n). • P(5), P(7) and P(11) are all true. • P(2) and P(4) are false. • For all integers k ≥ 6, if P(k) is true then P(k+ 1) is true. What is the smallest integer x for which you can be sure that P(n) is true for all integers n ≥ x? Answer:
Provide a combinatorial proof (argue that both sides count the same thing) for the following: 1+2+...+x = ((x+1) choose 2) Note: This is usually proven by induction but please refrain from using any induction in this proof. Hint: The above equation is also written as 1+2+...+x = (x(x+1))/2
Your answer MUST have a RIGOROUS proof (not just explanation, some math that THOROUGHLY demonstrates ALL cases work) AND Farey Sequences