Which of the following statements is (are) true for all positive integers? 1) n' + 2n is divisible by 3. 2) 5* – 1 is divisible by 24. 2x

Q: (b) Prove by induction that (3i – 2)² = n(6n² – 3n – 1) for n 21 %3D i=1

A: Prove the equation ∑i=1n3i-22=12n6n2-3n-1 for all n≥1. Prove the equation for n=1. Substitute n=1…

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: Give a formula for the nth positive even integer: Here is a theorem: 1+ 2+3 + ... + n п(n+1) n = Use…

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Q: 4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1

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Q: 5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.

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Q: 8.) Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers. 9.) Is 9"…

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Q: 5. Prove by induction that for positive integers n, E-1(3i² + 2i + 5) = ÷(2n² + 5n² + 13n).

A: Given statement is : ∑i=1n(3i2+2i+5)=12(2n3+5n2+13n) Step1: Let us check if or not the given…

Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

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Q: Prove by mathematical induction that 3n+2n+3 is divisible by 4 for all positive integers

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Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n>=1 in question number (7) Sum of squares of first n natural numbers…

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n k +1 j=k

A: Recall that KCk+1=0. We will use it in the proof.

Q: IM: Prove that n(n+1)(2n + 1) %3D 6.

A: We prove the given statement by using the sum of first n natural numbers.

Q: Write a short proof of the following and give a counterexample If n is an integer, then 2n + 1 is…

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Q: 3. Find integers s and t so that 1 7s + 11t. Show that they are not unique.

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

Q: (b) Prove by induction for all n >1 that an – bn = 2bn-1· %3D

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Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

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Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: Prove that 10 134n+1 + 115n+2 – 4 for all n E N.

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Q: 8. The identity element of negative integers (Z-) under the operation * defined as a*b = min(a,b) is…

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Q: 13. Prove by Math Induction e* > 1 + x for all positive integers x > 1

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Q: Use the permutation and combination formulas to prove the following. (a) For all integers n 2 2,…

A: We will solve the problem

Q: 17. 1+3n 0.

A: Alogrithm: Obtained P(n) and understand it's meaning. Prove that statement P(1) is true. Assume…

Q: 4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and…

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Q: 9. (a) Prove: If y>0, then there exists ne N such that n-1s y<n. ☆ (b) Prove that the n in part (a)…

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Q: Suppose that k and n > k are fixed positive integer. Justify the identity n n + 1 %3D k +1, j=k

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Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: (a) Prove that for all n 2 0, Eo fi = fn+2-1 where fi-1 is the th Fibonacci number.

A: Fibonacci sequence

Q: Is the following statement true or false? For every odd integer n, To answer this question, let n be…

A: Let, n be any odd integer. By definition of odd, n=2k+12, for some integer k So, by solution,…

Q: Determine whether 5 and 7 are irreducible or not in Zi]. O 5 is reducible but 7 is irreducible. O 5…

A: We have to check

Q: 1. Prove the following using direct proof. a. If 5x – 3 = x + 5, then x = 2. b. If x and y are both…

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Q: 9) +÷+÷+ 2" -1 for positive integers n. %3D .. 2n

A: Given: 9) To prove 12+14+18+... + 12n = 2n-12n for positive integers n.

Q: 2) If n is an odd integer, then H= (n-1) 2

A: To prove that n2=n-12 such that n is an odd integer.

Q: 3. Find integers s and t so that 1 7s+ 11t. Show that they are not unique. %3D

A: Apply the division algorithm, we get 11 = 7*1 + 4 7 = 4*1 + 3 4 = 3*1 + 1 Hence, 1 = 4 - 3*1   = 4 -…

Q: Verify by mathematical induction Ž (2i – 1)° = n°(2n² – 1) for alln2 1. - i=1

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Q: Determine whether 5 and 7 are irreducible or not in Zi. O 5 is reducible but 7 is irreducible. O 5…

A: Here , both 5 and 7 are irreducible because both cannot be reduced further. As both are neither in…

Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: Let, a1 = 3 and for n > 2, an = 2an-1 +1, express an in terms of n. %3D

A: We will find out the required expression in the term of n.

Q: tn be a positive integer and M = [2 01 %3D en M" 2n equal to the above None of the mentioned

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Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

Q: Prove the following using the appropriate method.    Let a, b be integers. If ab is even, then at…

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Q: 1.Then for positive integer n, A" is 0 1 Let A = 0 n² In 1 II In 0 1 II 0 0 2n In-1 n-1 2.

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Q: (2) (a) Prove by contrapositive : Va, b e R* : a <b → a < a +b < b. (b) Prove by contradiction : For…

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Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: give direct proof by contradiction that is 5n-1 is odd, then n is even

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Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

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Q: 2. Prove that n² > 2n for n >2

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