1. Prove that nº + n is even for every integer n.

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Prove thq+ for all integer a, b, and c if 9+b of a and k is even.

A: Introduction: Even and odd numbers are also referred to as: Even Number: An even number is one that…

Q: 1. Let n be an integer. Prove by contradiction is even.

A: Given that n2+5 is odd. If possible let n is odd. Then we get, n = 2k+1

Q: Prove that for every non-negative integer n the value of n² + n+41 is prime ?

A: The number n² + n + 41 is not prime for every natural number n. Let, f(n) = n² + n + 41

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.

Q: Prove that the statement is true for every positive integer n.n < 2n

A: n<2n Let us prove this statement by using the Principle of Mathematical Induction. When n=1,…

Q: 1 Prove that + n is an integer for any n E N. 6

A:

Q: Prove: for every positive integer n, 2"> n

A: As per the guidelines, since there are more than one question only one question will be answered.…

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

A:

Q: () is even. 3. Prove that for all positive integers n, the number

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Q: n Σ k (k!) = (n+ 1)! - 1. %3D k=1

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Q: -ん 2 kodd

A:

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n k +1 j=k

A: Recall that KCk+1=0. We will use it in the proof.

Q: Prove that 8 | (54" +7) for every positive integer n.

A:

Q: Prove that the statements here are true for every positive integer n.   1 · 2 + 2 · 3 + 3 · 4 + · ·…

A: To prove: For every positive integer n, 1.2+2.3+3.4+.......+n(n+1)=n(n+1)(n+2)3

Q: 7) Prove that for any integer n, n' -n is even.

A: We need to prove that for any integer n, n3-n is even, Now, an integer can be either even or odd.…

Q: Let n be a positive integer. Prove that (:) n +2. +...+n. n2"-1.

A:

Q: snip

A: To Prove n! ≥ 2n for all intergers n≥4We may prove this by Induction Method.

Q: R.4. Prove that for every integer n, n³ is even if and only if n is even.

A:

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

A: Answer

Q: Prove that for all positive integers k < n, the equality n n n =2n-k k k i=k holds.

A:

Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

Q: Prove that I +:+ + for every positive integer n.

A: In this question, principle of mathematical induction is used. Principle of Mathematical Induction…

Q: Prove that the following equality is true for all integers n>1 [n(n + 1)] 13 + 23 + 33 + ·..+n³ 2

A:

Q: Suppose that k and n > k are fixed positive integers. Justify the identity k+1,

A: Given k and n≥k are fixed positive integers.

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: (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: that for any integer n, n -n is even.

A: Prove that for any integer n, n3-n is even.

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

A:

Q: Prove that for any integer n, n' -n is even.

A: CASE I When n is even Let n=2m,m∈N⇒n3−n=2m3−2m⇒2m4m2−1=2η,where η=m4m2−1Thus,n3−n is even when n is…

Q: Prove that for all integers n, if n² is even then n is even.

A:

Q: Prove that for every positive integer n, there are n consecutive composite integers.

A: This question is related to Descrete Mathematics

Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

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: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: Prove that the statement is true for every positive integer n.

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

Q: Prove the following: for all integers n > 5, n2 < 2"

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Q: 18. For every positive integer n, 2n < 2".

A:

Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

A:

Q: Prove that the statement is true for every positive integer n.

A: 13+23+33+---+n3=nn+122 Let us use the Principle of Mathematical Induction to prove the given…

Q: Suppose that k and n > k are fixed positive integer. Justify the identity 'n +1° k +1 k j=k

A: We have to solve given problem:

Q: 3n Prove that for every integer n > 0, 9|(4" + 8).

A: To prove: For every integer n≥0, 9|43n+8. Proof by induction method.

Q: Prove by contradiction that 6n + 5 is odd for all integers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Prove that the statement is true for every positive integer n.

A: a&gt;1, then an&gt;1

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.