a. We define the sequence of numbers an for 0 ≤ n ≤ 3 an−1+ an−2+ an−3+an-4 if n ≥ 4. Prove that an = 1(mod 3) for all n ≥ 0. Please show your working process for the proof.

Q: Prove this claim: For every positive integer d and every integer n, the following equation is…

A: Using division algorithm we solve this problem. The steps are given bellow.

Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

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Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

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Q: 2. Let N = 210. Prove that N-p is prime for every prime p such that N/2 < p < N. Find a prime number…

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Q: 62. Prove part (b) of Theorem 4.4 in the case when n is odd: If n(n+1) n is a positive odd integer,…

A: ∑k=1nk=n(n+1)2 , n is odd integer

Q: Determine the least upper bou You must fully justify your an

A: We have to find least upper bound of the given set

Q: For positive integers m > n , show that -2 + n- 1 -3 + ...+ n + n-1 n - 1 + in

A: The right-side of the given equation can be expressed as…

Q: п(п+1)(п+2) E}=1jG + 1) = а. 3

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Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

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Q: 3.5. Let n E N. Prove that if n + ÷ < 2, then n² + n2 <4.

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Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: (a) Let n be any integer. Prove that n2 is congruent to 0, 1 or 4 modulo 8.

A: Note: According to bartleby guidelines we have to answer only first question please upload the…

Q: Consider the sequence an = in + (-i). Even though the formula for a refers to complex numbers, all…

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Q: d. Prove that for all integer n ≥ 2. 3n > 2n + n²,

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Q: Show that for every integer n, either 3 | n or n² = 1 (mod 3).

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Q: Prove the following step by step with explanation please,

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A: Given that, a can be written as base 10 x= a0+10a1+ ......+10nar We know that, If a divides b…

Q: Show that for all n ≥ 1, the n ₁²>(n+¹) " +1\ in

A: We have to show that for all n⩾1 , the inequality                  ∏j=1njj⩾n+12n(n+1)2Since…

Q: are 51 software students taking Let pi, p2,.. Psi be their final cam scores. Hence, cach p, is an…

A: Given that p1, p2, p3,..., p51 are the final exam scores of 51 software students who are taking MATH…

Q: Let A = {0, 1}. Determine all relations on A

A: The numbers of relations on A  2n2 n is the number of elements in the set 222=24=16

Q: 4. Find the largest value of n such that 2"|16!.

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Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: Assume that 0  ≤ a1 ≤ 1 and that  an+1=3√(1+an)/2 for all n ≥ 1. Prove that (a)0  ≤ an ≤ 1 for all…

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Q: 1. Prove that f1 + f²+...+ ƒ/²/2 = ƒnƒn+1, where n is a positive integer and fn is the nth Fibonacci…

A: The Fibonacci numbers are the numbers in the following integer sequence.…

Q: Let F(n) denote the Fibonacci numbers. Prove the following property for n ≥ 3 directly from the…

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Q: 5. The Fibonacci numbers are defined as: fo = 0 , fi = 1 , and fn = fn-1 + fn-2 , for n > 2. Give a…

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Q: 2. Prove that n + 100 > 3" if n is a positive integer with 1 < n < 3. (Hint: Proof by cases)

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Q: Find all n € N such that 2n +1 < 2ª. Prove your answer by induction. Find all n E N such that n² <…

A: First we find all the natural number n for which the given expression is true. Then we prove the…

Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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

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Q: Inspired by Fibonacci, let us define a new series of numbers which we'll call the Conway numbers: f₁…

A: The new series defined as, f1=1f2=3fn=fn-1+fn-2, for n≥3. Thus we have the sequence 1, 3, 4, 7, 11,…

Q: p, q, pq | (apq − ap − aq  + a) for all a.

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Q: inf{a,b, : n e N} > (inf{a, : n € N}) × (inf{b, : n E N}). (1) Does the inequality (1) continue to…

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Q: n(n+1) for all n EN, • Given that 1+ 2 +3+..+n = the sum 70 + 71+72 + .+ 100 = 2

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Q: 18.) Compute fg (Note: each should only be one In 2₁, we write -1 as f(x)=x²-3x +2, term in the…

A: fg(x) = f(x)g(x)

Q: Define the Fibonacci numbers as Fo = 0, F₁ = 1 and for n ≥ 2, set Fn = Fn-1+Fn-2. Show that Fn+1Fn-1…

A: Let's prove the identity Fn+1Fn−1−Fn2=(−1)n for n≥1 using mathematical induction.Base Case (n =…

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

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