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

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: Show that if n is an integer and 3n3 + n is odd then n is odd, by using a) Proof by Contrapositive…

A: To prove i - if n is integer and 3n3+n is odd then n is odd a) By contrapositive :   (P→9)≡(~q)…

Q: Prove: If x, y, z are integers with x2 + y? = z², then either x or y (or both) is even. Hint: Do a…

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Q: Prove that 3"< n! whenever n is a positive integer greater than 6.

A: Here we are to prove 3n&lt;n ! whenever n is a positiveinteger greater than 6.To solve this problem…

Q: Show that for all positive integers n, 13 + 2° + ...+ n³ = (1+2+3+ ...+ n)?

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Q: Prove: For integers m, n, 12mn – 9 is an odd integer.

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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 Vn e N (the set of natural numbers), ÷+ ɛ N. 3 +

A: We have to use proper manipulation.

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Prove that 3n<n! if n is an integer greater than 6

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Q: Prove that if n is a positive integer then n° z 2 mod 9. Show all your argument. Hint: use proof by…

A: Euclid-division lemma : Suppose a and b are two positive integer and when a is divided by b, then…

Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n &gt; 0.

A: Let a be a non zero real number , where m and n are integers…

Q: Prove: If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

A: See solution below

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n&gt;=1 in question number (7) Sum of squares of first n natural numbers…

Q: Q prove that for ony natural number n,2+4 +2n=1/n+1) %3D

A: The solution is given below.

Q: Prove by using a direct proof that if n is an odd number, then n2 is also odd.

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Q: 7) Use a direct proof to show that: a) The sum of an even integer and an odd integer is odd. b) If n…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: Show that if n is an integer and 3n3 +n is odd then n is odd, by using a) Proof by Contrapositive b)…

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Q: Prove by contraposition: For all positive integers n, r and s, if rs ≤ n, then r ≤√n or s ≤√n

A: Consider the statements p and q as-                 p : For all positive integers n, r and s,   rs ≤…

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

A: Concept:

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

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Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

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Q: #3. Let n be an integer with nz1 and Let b, bz . bn be ... positive real numbers, not all egual,…

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Q: Let k be a positive integer. Prove ÷VE. Note: do not use induction. j=1

A: The solution are next step

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: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: (":')-(:)·(,".) n +1 n n - Prove that for all integers n > i > 1. i 1

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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: Let n be an integer. Prove that if 7 divides (n – 2) or 7 divides (n – 5), then 7 divides (n² + 3).

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Q: Q 2: Suppose r is a real number other than 1. Prove using mathematical induction that for every…

A: Proof by induction

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

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Q: Prove that 2n>n2 if n is an integer greater than 4.

A: According to the given information, it is required to prove the given statement.

Q: Let pn be nth prime. prove that the sum 1/p1+1/p2+…..+1/pn is never an integer.

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Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

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Q: Prove that n2 +1 > 2? when n is a positive integer with 1<n<4.

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Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.

A: Sol :- To prove:- 2n+3&lt;=2^n if n is an integer greater than 3 We prove this by induction For n=4…

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n V3n

A: For all positive integers n, we need to prove that, 12·34···2n-12n&lt;13n

Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

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: For any positive integer n and r = 1,2, ...,n, we have G) = (","') + C" - }). True O False

A: The expression nr can be presented in the factorial form as follows. nr=n!(n-r)!r! The expression…

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: If x is an odd integer and y is an even integer then x + y is odd. Prove using direct proof.

A: Given: x is an odd integer y is an even integer

Q: Prove that n² + 1> 2" when n is a positive integer with 1<n<4.

A: Given: is a positive integer with. To prove: .

Q: prove that n^2 +1 >= 2^n when is a positive integer with 1<=n<=4

A: To prove n^2 +1 &gt;= 2^n when is a positive integer with 1&lt;=n&lt;=4

Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

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