11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or n=4k+3, where in each case k is an integer.

Q: Let denote the n-th prime number. Show that Pn Pn n=1 diverges.

A: Divergence of the series

Q: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

Q: (5n)! 40"-n! is natural number for every natural number n. Prove that

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Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

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Q: 1 se mathematical induction to prove that 1 + + > Vn +1 for every integer n > 3. + ... Vn

A:

Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: Prove that 2n+1 > (n + 2) · sin(n) for all positive integers n.

A:

Q: 2. Let n be a positive integer. Show that 4n 4n-k (4n 3 4n k=2n (Hint: an argument similar to the…

A: Solve the question by first opening the summation on the left side of the inequality. Simplify the…

Q: Use induction to prove: for any integer n 2 0, 2. 3 = 3"+1 - 1. %3D j=0 Base case 2.3 = j= n = ,…

A: for any integer n≥0,∑j=0n2. 3j=3n+1-1

Q: 1.) Prove every integer n >= 1, 1+5+9...+(4n-3) = n(2n-1).

A: 1.Apply mathematical induction to prove the statement. Let the given statement be Pn:…

Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

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Q: The sequence {n/(n + 1)} has a least upper bound of 1 Show that if M is a number less than 1, then…

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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: Use induction to prove (2k – 1) = n² – 4 k=3 for all positive integers n2 3.

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Q: 6"- (-4)" %3D k 1 k odd IM- 2]

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Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

A: Given statement, The sum of any two integers of the form 4k + 1 is always……

Q: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

A: See the attachment

Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

Q: Prove n (2k – 1) = n² - k=1 for every n EN.

A:

Q: 1. Prove that the negative of any even integer is even. 2. Prove that the difference between an even…

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: (1.3) Show that n - n (n – k) . k + (k + 1) k k 1 - for any two positive integers k and n with k <…

A: Let's find.

Q: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

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Q: Show that Ekin d(k)3 = (Ekn d(k))2 for all positive integer n. %3D

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Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

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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: A sequence c0,c1,c2,.... is defined by letting c0=3 and ck=(ck-1)2 for every integer k > 1.  Show…

A: Let's write some initial terms of the sequence and observe the pattern.

Q: Suppose that fis defined recursively by: f(0) = 5 and f(n+1)=2 fn+5. Find A(1), A(2), (3) and A(4)?…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Show that 2n+1≤2^n  for all n≥3.

A:

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

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Q: 1. Prove that (fn+1fn+3 – fnfn+4)* = 4 for each positive integer n.

A: We need to prove this, fn+1fn+3-fnfn+42=4

Q: Prove n (2n = n n 1 k=1

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Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: Prove that if h> -1 then 1 +nh ≤ (1 + h)” for all non negative integers n

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Q: 5 = t(n + 1) = t(n +2) = t(n + 3) holds for n = 3655 and 4503. (a) Verify that t (n) (b) When n =…

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Q: о Prove 2 (2n – 1) = n n - п - 1 k=1

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Q: Prove that 2+1> (n + 2) · sin(n) for all positive integers n.

A: Concept: Principle of mathematical induction: For each natural number n, mathematical induction is a…

Q: Let n and k be integers with 1<k<n. Show that n n+1 Zk=1 (k - - n

A: Given Solution:

Q: 4. Show that P(n,k)= P(n – 1, k)+ k · P(n – 1,k – 1). Using Combinatorial Proof.

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Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

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Q: Prove that 11 is the only prime number of the form 4n2 - 25.

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Q: part ii and iii, thank you

A: According to pascal law,

Q: 11. If n is an odd integer, show that n + 4n² + 11 is of the form 16k.

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

Q: Prove that n^2+5 >=4n for all integers, n>0.

A:

Q: Prove using mathematical induction that for every nonnegative integer n. ¡(i + 1) +1 (If n = 0, the…

A: Answer

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: Prove that n^2 is an even number if and only if 3n + 1 is an odd number by proving  If n^2 is even…

A:

Q: Prove that for any positive integer n, there is a k e Z>o and ao, a1, . .. , ak E {0,1} such that k…

A: Please check later steps for detailed answer

Q: For n2 1. prove using mathematical induction: 2" s 2n+1 - 2n-1 - 1

A: We have to solve given problem:

Q: oof. Assume that n is an odd integer. Then n 2k + 1 for some integer k. Then %3D 7n - 5 = 7(2k+1) -…

A: Theorem If n be an odd integer the 7n-5 be an even integer.