(a) Find the product a b, given that a = b = √9-√√9-√9-√7. b) Find all sequences a1, a2, a3,....of real numbers such that for all positive integers m, n ≥ 1, we have V6+√6+√6+√... 5+ √√6 + √6. am+n = Amn = am + an m²an+n²am + 2aman. - mn.

Q: Let n denote an integer. Prove that 5|n and 7|n if and only if 35|n.

A:

Q: Prove n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

A: We have to proff the question

Q: (a) For every integer n, the number n² — n must be even. (b) For all real numbers a, a² < a¹. (c)…

A:

Q: Show that 7 divides 29(52k) + 17(25k-2) for any integer k ≥ 1.

A:

Q: greatest common divisor(g.c.d)

A:

Q: Assume an unlimited supply of coins worth 5 cents and 9 cents. Prove that some

A:

Q: 7. (20 points) Prove by induction that, for every positive integer n, we have n· (5n + 11) (3 + 5i)…

A:

Q: 19 (e) a = 1575, b = -231 (a) 3710

A:

Q: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

Q: (a) Prove that for every integer n ≥ 2, 1 n+2 1 + n+1 +...+ 2n

A: Since you have posted a multiple question according to guildlines I will solve first (a)question for…

Q: For integers a and b, prove that (a + b) 3 a3 + b3 (mod 3).

A:

Q: ) 6. Express the sum 9 + 28 + 65 + ... + (n'+1) using E-notation.

A: We have to find sum of series:

Q: Find the number of ordered pairs (m,n) such that m and n are positive integers in the set {1, 2,…

A: Let us consider Tm=2m+1.  m Tm 1 3 2 5 3 9 4 17 5 33 6 65 7 129 8 257

Q: Suppose a and b are positive real numbers and n is a natural number with n 2 2. (a) Use the…

A: Let a and b are positive real numbers and n is a natural number with n≥2. Use the difference of…

Q: Suppose, and no are the smallest of two positive integers so that 300x is a perfect square and 300y…

A: The given problem is to find the values of x and y and sum of value of x+y for which smallest…

Q: Use induction to prove that 3 | (2²n — 1) for all nonnegative integers n.

A:

Q: (a) Let n EN. Determine the set of all integers which leave a remainder n+1, n+2, n+3 when divided…

A: PART A As it is given that 2n+1, 2n+3, 2n+5 are consecutive odd integers so they are pairwise…

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

A: Given that a and b are natural numbers and  a , b ≥ 2 .  Also, given that gcda,b = 1 and ab = x+yn…

Q: Please help with both (a) and (b) in the picture.

A: Please refer the image below

Q: Show that for every positive

A:

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

Q: Find an explicit formula for a sequence ao, a1, a2, a3, . . . that satisfies — 24. (Hint: ат Cr" +…

A: To find the explicit formula for a sequence given using recurrence relation find the characteristic…

Q: Let p be a prime number. Prove that (p -1)! a power p + (3p power2  +1)a is divisible by p, for any…

A: Consider the provided question, (a)

Q: Euler's formula for generating amicable pairs of integers is the following rule: = Suppose n> m > 1…

A: Note : In the Euler's formula for generating amicable pairs it is r = 2n+m (2n-m + 1)2 - 1 and not…

Q: Use the proof by contradiction to show that there are no three different natural numbers so that…

A: Use the proof by contradiction to show that there are no three different natural numbers so that…

Q: For the pair of integers 315 and 825: (a) Find the prime factorization of each number. (b) Find the…

A: As per our company guidelines we are supposed to answer only first three sub-parts. Kindly repost…

Q: Prove by mathematical induction that for all integers n 2 0: a 5" + 2 x 11" is divisible by 3, b 33n…

A: (a) Case 1 : n=1 51+2×111 = 5+2×11= 5+22= 27 As 27 is a multiple of 3 , hence the result holds for…

Q: Define a recursive sequence of complex numebers as,  g0 = 1, g2 = v,..., gn = u gn-2 +v gn-1, where…

A:

Q: p, q, pq | (apq − ap − aq  + a) for all a.

A:

Q: Show that the polynomial x + x

A:

Q: Use the method of finite differences to find a closed polynomial formula for the following sequence…

A: 1, 3, 21, 91, 273, 651, 1333, 2451, 4161, 6643,... find the first difference D1=2,18, 70, 182,…

Q: For each pair of integers r and s below, Find the greatest common divisor d. Then, find m, n, such…

A: To determine the greatest common divisor, d and then to find m and n such that d = mr + ns. (a) r=32…

Q: _4n + 47). Show that there are infinitely many positive integers n that satisfy the division 23|(n²…

A:

Q: (6) Find the sum of the integers for 7 to 508 Tmelusive,

A: first term (a0) = 7last term( an) = 508,     d = 1 ( natural no.)first we have to find total no of…