* 30. Let n and k be integers with 1 ≤ k ≤n. Show that n Σ() ( k=1 k n - 1) = (2n+2)/2 - (2n).

Q: n 2n + 2 /2 n 2n k k=1 \k –1, п+1 n -

A: The objective is to prove that ∑k=1nnknk-1=2n+2n+1/2-2nn This theorem can be proved by induction,…

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: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

A: See the attachment

Q: e3x e(n+1)x = 1+ e" + 2 enx Sx 6. n! (n + 1)!

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Q: Verify that the following inequality is true for all integers n > 1 1 1.3.5. .... (2n 1) - 2n…

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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: 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: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

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Q: (2n- 2. Give a combinatorial argument to show that k- ways the number of ways to select a committee…

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Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: 81. Prove: 2+4 +6+ + 2n = n(n + 1). ...

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Q: Prove that 6 divides n³ - n whenever n is a nonnegative integer.

A: In this problem we have to prove that  divides  where  is a nonnegative integer.

Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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Q: For each statement below decide if it is TRUE or FALSE. • There are two integers n and m such that…

A: We will find out whether the given statements are true or false.

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: 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: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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

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Q: 5. Prove that p(mk) = mk-1p(m) for all positive integers m and k.

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Q: Let n = pp2 Z+ where p₁ and p2 are distinct prime numbers and T1, T21. If o(n) = 60, then determine…

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Q: 10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime.

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Q: Show that yn for all n=1, 2₁- ✓

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Q: 5. Let m be the smallest positive integer such that m² + (m + 1)² ++ (m + 10)² is the square of a…

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Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

A: The given problem is to prove the given following identity that sum of squares of first n natural…

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

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Q: 8. Let K be an integer that is greater than 1, a perfect square, and equal to Ei for some integer D.…

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Q: Prove that if n = nxn-1 holds integer value of -n, which means that for a positive integer value n,…

A: We have given that ddxxn=nxn-1 for any positive integer n. We need to show that, ddxx-n=-nx-n-1. We…

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

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Q: Prove that 31 (n³ +5n+6) for every integer n ≥ 0.

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Q: - Prove that for every positive integer 11, 1.2.3+2.3.4+... +n(n+1) (n + 2) = n(n+1)(n+ 2)(n+3)/4.

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Q: It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2)…

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