4. Prove thatC(k, k) + C(k+1, k) + · · + C(n, k) = C(n + 1, k+1) for 0 ≤k≤n(Hint: Use induction on n for a fixed arbitrary integer k, as well as Pascal's Formula)

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Answered: 4. Prove that C(k, k) + C(k+1, k) + · ·…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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28. Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0, 0) E S. Recursive step: If (a, b) € S, then (a +2, b + 3) € S and (a +3, b+2) E S. a) List the elements of S produced by the first five applications of the recursive definition. b) Use strong induction on the number of applications of the recursive step of the definition to show that 51a + b when (a, b) E S. c) Use structural induction to show that 51a + b when (a, b) E S.
2. Prove the following, using induction: (a) If an+1 = an/(an +1) prove that an = ao/(nao+1) for all n ≥ 0. (b) Define fo = 0, f₁ = 1 and fn = fn-1 + fn-2 for n ≥ 2. Show that 3 fak for all k ≥ 0. (c) Show that for 2" > 4n for n > 5. (d) Show that 3 | 5 - 2" for all n. (e) Show that every n ≥ 8 can be written as 3a+5b for some a, b € N.
4. Let P(n) be the statement that a postage of ŉ cents can be formed using just 4-cent stamps and 7-cent stamps. The parts Page 363 of this exercise outline a strong induction proof that P(n) is true for all integers n ≥ 18. a) Show that the statements P(18), P(19), P(20), and P(21) are true, completing the basis step of a proof by strong induction that P(n) is true for all integers n ≥ 18. b) What is the inductive hypothesis of a proof by strong induction that P(n) is true for all integers n ≥ 18? c) What do you need to prove in the inductive step of a proof that P(n) is true for all integers n ≥ 18? d) Complete the inductive step for k ≥ 21. e) Explain why these steps show that P(n) is true for all integers n ≥ 18.
Prove by weak induction on n: IMI Li ₂² n(n + 1)(2n + 1) 6
A bar of chocolate consists of n ≥ 2 connected pieces. A component is a portion of the originalbar consisting of fewer than n pieces. A component consisting of only 1 piece may not be dividedfurther. A break is the act of splitting the bar or a component of the bar into exactly 2 smallercomponents. For instance, the 1st break separates the bar into 2 components each containing fewer than npieces. Use strong induction to show that n − 1 breaks are required to separate all of the chocolatepieces from each other.
2. show work
The Fibonacci numbers fi, f2, f3,... are defined by setting f₁ = f2 = 1 and fn = fn-1+ fn-2 for all integers n ≥ 3. (a) Prove that for all integers n ≥ 3, gcd(fn, fn-1) = gcd (fn-1, fn-2). (b) Prove by induction on n that ged(fn, fn-1) = 1 for all integers n > 2. n (c) Prove by induction on n that f = fn+1fn for all integers n ≥ 1. i=1
5. Use ordinary induction to prove that for every positive integer n, n-n is a multiple of 6. Only proofs by induction are accepted.
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Use induction to prove that for all n € N, 1 + 3 x 5 1 1 x 3 + 1 + 5 x 7 + 1 (2n − 1)(2n +1) = n 2n + 1
10) Identify the mistake in the following “proof" by mathematical induction. (Fake) Theorem: For any positive integer n, all the numbers in a set of n numbers are equal to each other. Proof (by mathematical induction): It is obviously true that all the numbers in a set consisting of just one number are equal to each other, so the basis step is true. For the induction step, let A = {a1, a2, a3, . .., an41} be any set of n +1 numbers. Form two subsets each of size n: B = {a1,a2, A3, . numbers in A except an+1, and C consists of all the numbers in A except a2.) By the induction hypothesis, all the numbers in B equal a1 and all the numbers in C equal a1, since both sets have only n numbers. But every number in A is in B or C, so all the numbers in A equal a1; thus all are equal to each other. .., a,} and C = {a1, az, a4, . .. , an+1}. (Precisely, B consists of all the

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4. Prove that C(k, k) + C(k+1, k) + · · + C(n, k) = C(n + 1, k+1) for 0 ≤k≤n (Hint: Use induction on n for a fixed arbitrary integer k, as well as Pascal's Formula)