21. Prove that for all natural numbers n, ++ + n(n+1) n+1 a. Show it is true for n = 1. Show your work here: b. Assume it is true for n= k. Show your work here: c. Prove it is true for k + 1. Show your work here:

Q: Which of the following rules is used to find the number of positive integers less than 1000 that are…

A: To Determine: which of the following rules is used to find the number of positive integers less than…

Q: We know that if n is odd, n = 2k + 1 for some integer k. We also know that if n is even, n = 2k for…

A:

Q: The inductive step of an inductive proof shows that for k ≥ 4, if 2k > 3k, then 2k+1 ≥ 3(k+1). In…

A: We will check the validity of all given steps, then we can decide in which step we used the given…

Q: Prove the following: There exist no integers a and b for which 14a + 4b = 1. Please explain the…

A:

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

A:

Q: Find the largest integer k for which fl(19+2-k) > fl(19) in double precision floating point…

A: Advance maths

Q: Theorem: The number of diagonals of a regular n- Define P (n) = n(n-3) 2 gon is n(n − 3) 2 - for n ≥…

A:

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: c.) There is a conjecture called "Goldbach's weak conjecture" (yes, really!) that every odd number…

A: Given: Odd numbers greater than 5. To prove: Every odd number greater than 5 can be written sum of…

Q: Prove the property of the Fibonacci numbers, F(n) = 5(n 4) + 3F(n-5), for n ≥6

A: We have to prove the property of the given Fibonacci numbers F(n) = 5F(n-4) + 3F(n-5) ,  for n≥6

Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

A:

Q: ite P(0). Is P(0) true? Write P(k). Write P(k + 1). In a proof by mathematical induction that this…

A:

Q: The following is a valid identity for the natural numbers, true or false? n(n +1)(n +2) P(n):13 +23…

A: The solution is given as

Q: Suppose a is a positive integer. Simplify the expression a Σ((n + 2)³ + n³ - 2(n+1)³). n=1 [Explain…

A:

Q: The integers from 1 to 7 are written in order clockwise around a circle. I start with a counter on…

A: State SpaceTransition Probabilities

Q: Prove that 8 divides 9" – 1 whenever n is a natural number. For each natural number n, let P(n) be…

A: To prove that  8 divides 9^n-1 whenever n is a natural