Section 3.1 Homework 13 +23+... +n = 1n (n +1) for all natural numbers n. 1. 1 + ... + 1 1 1 + for all natural + 2. n(n+1) (n+1) 1(2) (2)3 3(4) 1-r* n 1 and any ne N for any r 3. Show thatr" 1-r k=0 1+2+22 +... + 2"- = 2" -1 for all natural numbers n. 4. 52 -1 is a multiple of 8 for all natural numbers n 5. 9" - 4" is a multiple of 5 for all natural numbers n 6. Use induction to prove Bernoulli's inequality: If 1 + x > 0, the 7. 1 Prove the Principle of Strong Induction: Let P(n) be a statement that is either true or false for each na Then P(n) is true for all n, provided that (a) P(1) is true (b) For each natural number k, if P(i) is true for all integers j s 8. 7444/ )2

Q: Use mathematical induction to prove that the conjecture P(n) below is true for n e Z* n(n + 1) E (21…

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Q: Prove n n(n + 1)(2n + 1) Σε ん? 6 k=1 for every n E N.

A: Given ∑k2k=1n=nn+12n+16, n∈N

Q: Chapter 4: 4.6.8, 4.6.9 (a),(b), 4.6.11 (b), (c), 4.6.14 1. Prove that for any natural n the…

A: We can prove this by induction method

Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

Q: Prove that (1² × 2!) + (2² × 3!) + …· + (n² × (n + 1)!) – 2 = (n+ 2)! (n – 1) ... For all n E N

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Q: E. Prove by Mathematical Induction that for any natural number n 1+5+9+13 + .. .+ (4n – 3) = n(2n –…

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Q: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

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Q: Prove that for every n E Z, n² – n is even. (Proof by cases)

A: In the given question we have to prove that for every n∈ℤ, n2-n is even.

Q: Assume that S = Σ(-1)"-¹. 999 9 99 8 O 1000 10 1 (n + 1)5 n=1 n=1 Find the smallest integer N such…

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Q: 1 Prove that + n is an integer for any n E N. 6

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Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

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Q: Prove the following by Mathematical Induction 1. 1/1(2) + 1/2(3) + 1/3(4) +...

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Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: 5. Use induction to prove that for every n E N we have E(k · k!) = (n + 1)! – 1 k=1

A: We want to prove ∑k=1n(k·k!) = (n+1)! - 1

Q: Prove that 12- 22 + 32+ +(-1)"n = (-1)**'n(n+1)/2 for all n E N.

A: Given : Prove that  12−22+32+......+(−1)n+1n2=(−1)n+1n(n+1)2   for n∈ℕ

Q: Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

A: We know that P(Ai) = 1 for i ≥1 Considering P(Ai), i=1,2,3,... to be independent events

Q: Prove the following using mathematical induction. 1. 1 + 2' + 3' + ... + n = n'(n + 1)2 4 2.

A: In this question we have to prove the expression using mathematical expression. Note: As per our…

Q: 2. Use mathematical induction to prove that the statements are true for every positive integer n.…

A: As per our guideline we are supposed to answer first three subparts.kindly repost other question.…

Q: Prove the following formula for all n E N by induction: 1+ 2·2 + ... +n· 2"-l = 1 + (n – 1)2".

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Q: Use induction to prove that n п(п+1) Σ- k for all n e N. 2 k=0

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Q: 3. Prove that if a > 1, then a"/n → o.

A: We used binomial inequality

Q: 1. Prove that if n E N, then E, k(k + 1) = n(n+1)(n+2). 3

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Q: 1. Prove that 12 + 22 + ...+ n² : n(n + 1)(2n + 1) %3D 6. for n e N.

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Q: 7. Prove by induction that E,(-1)² = (-1)"n(n + 1)/2 for all n > 1.

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Q: 9. (Chapter 10, #20) Prove that for every n E N, (1+2+3+ · ..+n)2 = 13 + 23 + 33 + ...+ n³.

A: To prove the result ''∀n∈N,1+2+3+...+n2=13+23+33+...+n3'' by Mathematical induction: First prove for…

Q: or every integer n ∈ ℕ, it follows that 1^2, 2^2, 3^2, 4^2 + ...+ n^2 = (n(n+1)(2n+1)/)/6.  P

A: Claim: 12+22+32+…+n2=nn+12n+16 Proof: Now k-13=k3-3k2+3k-1 Rearranging the above equation. we get,…

Q: السؤال 2 For all n E N, n> 1, there exists a prime p such that p|n. ylan O İhi O

A: True

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: Theorem 24. If ɛ > 0, then there is an integer N EN such that 0 < + < e. .

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Q: Prove by Mathematical Induction. n(n + 1)(2n + 1) k² 6 k=1

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Q: 1. Prove by induction that for all positive integers n, we have: 1+3+5+ + (2n 1) n2

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Q: Prove by induction  that the sum of the first n even integers starting at 2 is n(n+1).  (That is,…

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Q: Suppose that P(A;) = 1 for all i = 1, 2, .... Show that P(N A;) = 1.

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Q: For all n E N prove that 1 3 2n – 1 1 - 2 4 2n V2n + 1

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Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n n

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Q: Prove that, for all integers n > 1, 1+ 3+ 5+7+ ...+ 2n – 1 = n².

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Q: Show by mathematical induction, that for all n 2 1 n! > 2n-1

A: To show → by mathematical induction that                       V n ≥ I  , n I ≥ 2n-1Proof →Let ,…

Q: #1

A: We prove this by mathematical induction.To prove:

Q: Show that if M is a number less than 1, then the terms of {n/(n + 1)} eventually exceed M. That is,…

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Q: (5a) Let A = {3, 4} be a subset of S = {1,2,... ,6} and let n E S. || n2(n+1)² Prove that if is…

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Q: Σ- n+ 1) (2n + 1), n2 1 i=1

A: Induction method says that if n=1is true then assume that n=k then prove that the statement is true…

Q: 1. Use induction to prove that Σ m(n+1) for all n eN. k=0

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Q: Prove that for every n E N is a² | (a + 1)^+1. - 1)n+¹ na — (a + 1), a € N. - -

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Q: Prove the following equation for every n>=2 by mathematical induction.…

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Q: Prove the following using mathematical induction. Šr- n(n+1)(2n+ 6. 1. for all neN.

A: Question 1 solved ,pls ask other question in next turn

Q: Prove the following fact by induction: n For all n > 1, E(2j – 1) = n² j=1

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Q: Prove by induction that for all n ∈ N with n ≥ 8, 2^n ≥ 3n^2 + 5

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Q: Prove that for any natural n the following equality holds (n – 1)n(n+ 1) 1.2+2.3+...+(n – 1) · n =

A: We have to prove that for any natural n,            1⋅2+2⋅3+......+n−1⋅n=n−1nn+13. We will prove…

Q: 8

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.