Let i be a positive integer and A; = [1,i+1[. Then U2, A= O [-1,1] O R O 11, [ O [1, [ None of these

Q: Prove that for all positive integers n, we have 6″ – (-4)″ Σ (1) 5k 2 k−1 k odd =

A: Expand 1+xn 1+xn=C0n(1)nx0+C1n(1)n-1x1+C2n(1)n-2x2+.......+Cnn(1)0xn=∑r=0nCrnxn Expand 1-xn…

Q: Let i be a positive integer and A =] - i+1,1]. Then N1 A O 10,1[ O 10,11 O 10,1] O None of these

A: ]0, 1] is correct

Q: 14. Let (a, b) = 1 and 2 = y for some integers x and y. Then prove that there exists an integer n…

A:

Q: 3. Prove that for all x > 0 and all positive integers n, x3 et >1+x + 2! 3! n!

A:

Q: Prove that, nonetheless, for all integers a, if 2 | a and 3 | a, then 6 | a

A:

Q: Given a positive integer n, let a2,n–1 a2,n a3,n-2 a3,n-1 a3,n An = аn-1,2 аn-1,n-2 аn-1,п-1 @n-1,п…

A: Given: Here the given positive integer n, Let An = 00...00a1,n00...0a2,…

Q: that a

A: Introduction: Notation for dividing A by B. In other words, if a and b are integers, a divides b if…

Q: let b be a nonzero integer. prove that gcd (0,b) = |b|.

A: We need to prove that for a nonzero integer b,

Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

A:

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

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: (a) Let m and n be integers greater than 1. What is the order of the element (1,1) in Z/mZ x Z/nZ?…

A:

Q: Prove that o – 1 is divisible by 4.

A: Even numbers are numbers divisible by 2. Odd numbers are numbers that are not divisible by 2. The…

Q: Let i be a positive integer and A; =[-i+1,1]. Then U A;= %3D O 1-0,1[ O [-1,1] O R O 1-0,1] O None…

A:

Q: Let a, b € Z with a | b. Prove that an | bn for every positive integer n.

A: Given that           a, b∈ℤ with a|b. To prove an|bn  , we shall use mathematical induction.

Q: Prove that for all the integers a and n, if a | n, then a^2 | n^2

A: Given that a | n. This means n=ak, where k=integer. Then we square both sides: n2=(ak)2 or n2=a2k2

Q: Prove that for all positive integers k and n, with k < n, (:) - () - (^) n k 1 k n 1 k k – 1 k – 1 k…

A: 1+kk-1+K+1k-1+   ... ...+n-1k-1 add and subtract 1 and write 1 as a combination of k objects taking…

Q: 26. Let E be the set of all even integers, and let O be the set of all odd integers. Let X = {n €Z|n…

A: If y is an even integer then it will be of type 2n for some n in integer If y is an odd integer then…

Q: Prove that if a and b are integers of the form 4n+1, then ab is of the form 4n+1.

A:

Q: Prove, for all integers x, that if x2 is odd, then x is odd.

A: Result: Every odd integer n can be written as n=2k+1, where k is an integer. Every even integer n…

Q: 1. Prove that for every positive integer n there exist positive integers a11, a21, a22, a31, a32,…

A:

Q: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b € N, and odd c€ N such…

A: Since n is even and non-zero . Since n is non zero ⇒ n is either positive or negative. Let's write…

Q: Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a…

A: Given: Let k and n be two positive integers in that, we assume if n is odd.

Q: (f) Prove by induction that for any positive integer n, n(n +1)(2n + 1) j=1

A:

Q: Prove that for any positive integers a₁, a2,..., an, there exists integers 1, 2,...,n, such that a₁₁…

A: To Prove: for any integers a1,a2,...,an there exists integers x1,x2,...,xn, such that…

Q: a. Prove that for every integer a, if a³ is even then a is even. b. Prove that 2 is irrational.

A:

Q: 2n 2n 1/2n + 2 1, 2 n +1 n -

A:

Q: prove that if a and b are both odd integers, then ab+1 is an even integer

A:

Q: Prove that, for all integers n > 1, 1+ 3+ 5+7+ ...+ 2n – 1 = n².

A:

Q: Give an outline for a proof of "for any integer r and any positive integer d, if d|(x– 1), then d…

A:

Q: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b e N, and odd c e N…

A: Consider the given condition,  n∈ℤis a non-zero and even. The objective is to prove that there exist…

Q: Recall that a is a type 1 integer if there exists an integer k such that r = 3k + 1. Prove that if a…

A: x is a type 1 integer. So, x = 3k+1, for some integer k. So, x² = (3k+1)²

Q: Let z, y, z be integers. Prove that if z +y + zis odd, at least one of z, y, or z is odd.

A: We prove this by contradiction

Q: Let a be any positive number. Prove that |x| &gt;a if and only if x &gt; a or x &lt; -a.

A: Suppose

Q: Prove: If Am = O for some positive integer m, then I − A is nonsingular.

A:

Q: Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

A: To prove the divisibility property under the given conditions

Q: 2) Prove that o(g

A: To prove that, ogn∩ωgn is empty set.

Q: Let In = | (1– t²)" dt, for any nonnegative integer n. 1°) Prove that, for any positive integer n,…

A: Given:

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Q: Let x be positive integer. Prove If x2 + 1 is even, then x3 is odd

A: Consider the provided question, Let x be positive integer. We need to Prove that, If x2+1 is even,…

Q: Given a positive integer n, let a2,n a2,n–1 a3,n–2 a3,n–1 a3,n An = An-1,n аn-1,2 ап-1,n-2 аn-1,л-1…

A: For given a positive integer n, let An=00⋯00a1,n00⋯0a2, n-1a2,n00⋯a3,n-2a3,n-1a3, n⋮⋮⋱⋮⋮⋮0an-1,…

Q: Given positive integers k and m, there exists an integer N ≥ m such that

A:

Q: Let i be a positive integer and A; = [1,i+1[. Then U A OR O [-1,1] O [1, [ O 11, [ None of these

A: [1,inf[ is correct answer

Q: Let a, b, c be non-zero integers. Prove that gcd(a, bc) = 1 if and only if gcd(a, b) = 1 and gcd(a,…

A:

Q: prove For all integers a, b, c, and d, if a|c and b|d then ab|cd.

A:

Q: Let n be a positive integer. Prove that for all x € Z, if x" is odd, then x is odd.

A:

Q: Prove that 1 n i(i+ 1) n+1 i=1 whenever n is a positive integer.

A:

Q: Prove that for any positive integer n, Vn is either an integer or irrational.

A:

Q: Label each of the following statements as either true or false.     Let a and b be integers with a…

A: Given: a and b are the integerswith a=1=-0Formula: a|b…

Q: Prove that for all positive integers n, we have 6″ Σ (1) * _ "(-¹)" 2 k=1 k odd IG

A: