-1 (a) Determine the set of values of n for which n² +1 and 2 are positive integers. A 'Pythagorean triple' is a set of three positive integers a, b and c such that a² +b? = c². n² –1 and 2 n² +1 (b) Prove that, for the set of values of n found in part (a), the numbers n, 프큰 fom a Pythagorean triple.

Q: 2.) Let 1 > -1 be a real number. Prove that (1+1)" >1+ nx for all integers n > 1.

A: We use principal of mathematical induction (PMI) to prove this statement.

Q: Prove that 1– 2+2² – 2° + -.-+(-1)"2"=2**1 for all nonnegative integers n.

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

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Q: 4. For an integer n > 0, let hn denote the number of n-digit integers with two properties (1) each…

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Q: Show for all natural numbers n >= 3 that [(1 + (1/n))^n] < n.

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Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x &gt; – 1.

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Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: A. Prove that for all positive natural numbers n, 13 + 23 + . + n³ = n²-(n+1)²/4. + ...

A: Explanation of the answer is as follows

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: Let k, n be positive integers with k < n. Give a combinatorial proof of the identity (in the form…

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Q: For all integers 1. x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5.

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Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Prove by contradiction : For any odd integer n : n² – 7 is not divisible by 4.

A: Proving a statement by contradiction means assume the negation of statement true and prove the…

Q: -, 1+2+2 '+.+ 2 -1 for all natural numbers n.

A: We will use the principle of mathematical induction to prove the result.

Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

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Q: Use a combinatorial argument to prove that EC(n, k) = 2". k=0

A: The Multiplication Principle If one event can occur in m ways and a second event can occur in n ways…

Q: Let b a natural number and q1, q2, r1, r2 integers with 0 ≤ r1 < b and 0 ≤ r2 < b such that q1b + r1…

A: Natural Numbers: Natural numbers, that embody all positive integers from one to time, square measure…

Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

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Q: Prove that n- + 1> 2" when n is a positive integer with 1 <n<4.

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Q: A scientist noticed that the population of a certain country could be represented by the following…

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Q: - 5. Prove that an integer n is odd if and only if n° – 3 is even.

A: Answer

Q: Let ak = 4k + 2 and b, = (k – 1)3 + 3k + 3 for every integer k 2 0. What are the first five terms…

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Q: Use the definitions of even and odd to prove: For all integers n, n2+3n is even.

A: We have to prove, For all integers n, n2+3n is even.   We prove this by using Principal of…

Q: Prove that k (x)= - = n (n − 1) for integers n and k with 1 ≤ k ≤ n, using a: a) algebraic proof.

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Q: 4. Let n and k be two positive integers with n 2. Prove that (n 1)² divides n-1 if and only if n -1…

A: The given problem is to prove the given following statement. Given statement, let n and k be two…

Q: Prove that for any integer n if 3n+ 2 is even, then n is also even.

A: As per Bartleby answering guidelines if multiple questions are posted in a single picture then an…

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

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Q: Exercise 2. Use a combinatorial argument to prove É(:)-("*") n+r+ k=0

A: ∑k=0rn+kk=n+r+1r LHS=n0+n+11+n+22+⋯+n+rr       =n0+n+10+n+11+n+22+⋯+n+rr-n+10       =…

Q: 3. Prove by either Well Ordering Principle or induction that for all nonnegative integers,n: + 1) i3…

A: We prove by induction .Case n=1LHS=∑i3i=01=03+13=1RHS=11+122=222=1Induction hypothesis : Assume…

Q: Use combinatorial proof to show that 10 (2n - 5 3 n - 2 n | %3D 10 k 10 – k

A: Given:

Q: Prove that 3" < n! for all integers n 2 7.

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

A: Given Solution:

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

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Q: Prove that 1 + 3 + • • • + (2n - 1) = n2 for all natural numbers n.

A: Our Aim is to prove that 1 + 3 + .... + (2n - 1) = n2 for all natural numbers n.

Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: 5) Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4.…

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Q: Prove that: there exist infinitely many 5-tuples of practical numbers of the form (n-8, n – 4, n,…

A: It is given that 5-tuples of practical number of the form (n-8, n-4, n, n+4, n+8)

Q: Exercise 2.4.4: Showing a statement is true or false by direct proof or counterexample. i About…

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

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Q: (a) Show that 1++++ 1 21+, 2n-1 2n whenever n is a nonnegative integer.

A: We need to show that 1+12 + 13 + ..... + 12n-1 + 12n ≥ 1+n2We will show this by mathematical…

Q: Let's take an integer n to be odd if there is some integer k such that n = 2k – 1. Prove that the…

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

A: Answer

Q: 3. Prove that if n is odd, then 8 divides n2- 1. Prove that 6 divides n п. 4. Let n and k be two…

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Q: List the first 3 positive prime numbers d in Z so that the quadratic integers in Q[Vd] are precisely…

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Q: Prove that 1 2 + 2 · 3 + 3 · 4+ + n(n + 1) = "(a+1)(a+2) for every positive integer n. ... 3

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Q: Prove: 5" +9 < 6" for all integers n2 2.

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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: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: For all integers 1. x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5. Write a paragraph proof of…

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