(1) Prove that is a, b, and c are integers with c + 0 then ac = bc = a = b ( that is this a key step towards showing that fractions work as expected bec makes dividing by c looks possible for any c + 0.)

Q: 20 Term independent of x in expansion of x2 5 + 6x3 is? e 7

A: Here we have to find the term independent of x.

Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Prove that if n is a multiple of 4, then Fn is a multiple of 3.  

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Q: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

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Q: Prove by the principle of mathematical induction that 3n2 + 3n + 1 < 2n for n ≥ 8

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Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

Q: let k >= 2 show that n= 5^k-1 satisfies the equation 4σ(n) = 5n-1

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Q: Suppose n E Z. Prove that if n is even, then n2 + 5n + 4 is even.

A: Note: We know that the square of an even number is always an even number, since  Let x is an even…

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

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Q: 1. Complexity 2n – n³ + 2 Prove that O(n) 3n2 + 4n – 1

A: The big O notation is used to mention the important term r the biggest term in the equation. The…

Q: The formula for calculating the sum of all natural integers from 1 to n is well-known: n² + n 2 Sn =…

A: Given:  Sn= 1+2+3+.......................+n = n2+n2and,Qn=1.1+2.2+3.3+...............+n.n=n33+n22+n6

Q: 2. Prove by the Principle of Mathematical Induction that n°(n +1)² 1° + 2° + 33 + 4* +.. +n° = 4

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Q: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

Q: 12. Given: L1 = Z2; Z3 = 4 Zs 2 and 3 are complementary Prove: Ls 1 and 4 are complementary 2,

A: Two angles are said to be complementary when sum of their angles is equal to 90° or pie÷2

Q: n(2n – 1)(2n+1) 12+ 32 + 52 + ...+ (2n–1)² 3

A:

Q: The formula for calculating the sum of all natural integers from 1 to n is well-known: n² Sn…

A: This question is related to principal of Mathematical induction . solve in step 2.

Q: 4. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

A: Given- Let a1 = 1, a2 = 2, b1 = 0, b2 = 1 for n ≥ 3,an = 2an-1 + bn-1for n ≥ 2,bn = bn-1 + an-1 To…

Q: b) Prove that all numbers n > 20 can be represented as n = 4 * i +7*j for some non- negative i and j…

A: Prove that all numbers n≥ 20 can be represented as n = 4*i+7*j for some non- negative i and j by…

Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

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 Mathematical Induction to prove that for n > 1, n²(n+1)? No other method of proof is acceptable.…

A: We have to prove ∑i=1ni3=n2n+124 by using principal of mathematical induction. First check for  n=1…

Q: Prove that n3 + 2n is divisible by 3 for all n E N.

A: We prove that n3+2n is divisible by 3 for all natural numbers n.

Q: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: Prove T(n)= 2T(n-1)+n can be in 0(n2")using represented telescoping method

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Q: 5. Find and prove a formula for P(n,n-2) for n2 3.

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

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Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

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Q: Show that φ(n2) = nφ(n) for every n ∈ Z+.

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Q: Use any method to deter (a) 13 The series 8.

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Q: Prove that n² +n is divisible by 2, where n E N.

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Q: 2. Prove by the Principle of Mathematical Induction that 5-2 + 5-4 + 5-6 +... + 5-2n = 24-'(1 –…

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Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

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Q: The principle of mathematical induction can be formally represented as [P(1)V vk( P(k) → P(k + 1))]…

A: Q1.The principle of mathematical induction can be formally expressed as P1∧∀kPk→Pk+1→∀nPn. Hence,…

Q: 1. Use mathematical induction to show that n2 – 3n +4 is even for all n > 1.

A: To show that n2-3n+4 is even for all n≥1, using Mathematical induction.

Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

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Q: Use induction to prove that 16 divides 5n+2 + 13"+3 – 14 - for all n > 0.

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Q: Q.No.6. Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

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Q: Suppose X∼N(μ=2,σ=0.5)X∼N(μ=2,σ=0.5)  , then the z-score of x=2.7x=2.7   is choices: 1.4 1.1 1.6…

A: Given  X∼N(μ=2,σ=0.5) and x=2.7

Q: 6. Show that (") = 3() + 6n (5) +n³. Using Combinatorial Proof.

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Q: 2) use the mathematical induction to prove : A) 4" - 1 divided by 3 , Vn E N+

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Q: Show that the sum of the first n positive odd integers, 1 + 3 + 5 + .......+ (2n - 1), is n2.

A: Given: 1+3+5+...........+2n-1 is n2

Q: 6. Prove that for n> 2, 2(;) + (*) =n². п

A: We have too prove the given expression.   According to our guidelines we can solve only one question…

Q: In a Wilcoxon Mann Whitney test n1 is 9 and nprime is 7. What are the correct values for K1 and K2 ?…

A: In a Wilcoxon Mann Whitney test n1 is 9 and nprime is 7.  We need to find the correct values for K1…

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

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Q: Pls help quick

A: Introduction: According to the combination rule, if r distinct items are to be chosen from n items…

Q: 8. Working over R, suppose that M = PDP ' where P 1 Then, for any positive integer k, we have that…

A: To find Mk =?

Q: 14. Let a/b be a fraction in lowest terms with 0 <a/b<1. (a) Prove that there exists n e N such that…

A: