2. Ifr + 1, show that for any positive integer n, a + ar + ar +...+ ar" = a(rn+1 – 1) - r - 1 the Sooond D..

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

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Q: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

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Q: 2. Prove by mathematical induCHon that १) के) 22 32 42 2n is Hrue For all integers nz 2

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Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: 5n² is O(n³)

A: Suppose fn and gn are positive functions. We say that fn∈Ogn if there exist positive constants c, n0…

Q: Given that a =3 and 1 if a is even; a n+1 5а - 1 if a_is odd. Then tern number four, i.e., a а, is

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Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

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Q: Show that for n, m e Z+, gcd(2" – 1,2 – 1) = 28cd(n,m) – 1.

A: To prove : gcd(2n-1, 2m-1)=2gcd(n,m)-1where n,m ∈ℤ+

Q: Suppose n E Z. Prove that if n is even, then n2 + 5n + 4 is even.

A: Note: We know that the square of an even number is always an even number, since  Let x is an even…

Q: What is the first step to prove that S, 2+4 + 6+...+ 2n = n(2 + 1)? n(n + 1)? O Verify that S, is…

A: Here we have to determine the first step to prove that,  Sn⇒2+4+6+..........+2n=n(n+1)

Q: If "C, = K "C,, then K is n-1 %3D

A: Combination problem....

Q: If In z + In(x – 6) = In(5x), then r

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Q: Prove, by induction, that, for n2 5, the nth Fibonacci number satisfies Fn s 2".

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

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

Q: prove (1+ )" in <4 for all n21

A: In the given question we have to prove that  1+1nn&lt;4∀n≥1 . we can prove this by two methods one…

Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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Q: Prove NIn, +oo[= 0 ! n=1

A: To prove that ∩n=1∞[n,∞)=∅. Suppose that x∈∩n=1∞[n,∞). Then, x∈[1,∞), x∈[2,∞),.... That is, x≥n,…

Q: If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that

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Q: 6.20. For which integers n > 2 does there exist a o e A, such that |o|> n?

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Q: Suppose that A ¥BSCN D and x E A. Use the method of "proof by contradiction" to show that if x e C,…

A: Given data : x∈Aandx∉C To show : x∈B

Q: 1. Use PMI to show that for any n E N, 4 divides 5" - 1.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: For all NEZ, itn is odd and a?tb' for same adbt s", thenn mad 4=1.

A: here we can use even odd condition to solve this question.

Q: *prove that for . P+2+3+... +n° = 2 n=!,2,3,

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Q: We need to attempt to find an exact value of n such that  P(r ≥ 6) = 0.98. Clearly to guarantee…

A: We have to fill blanks from given statement Clearly to guarantee at least six people are literate,…

Q: Prove that the nth Fibonacci number fn≤αn−1, where α is the golden ratio.

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Q: 12 + 22,., +n? =n(n+ 1)(2n + 1), Vn eN

A: We will prove this as following.

Q: Give a direct proof of the claim: V n E N, Vk E N, if the last k digits ofn are divisible by 2k then…

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Q: 6- The special sotution 5014tion to the र१पच्दं०n . -ya)=x"e? 1 XP A/ Xex Ğ d/other than that

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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: If n is an integer greater than 2, then for all integers m, n does not divide m or n + m # nm.

A: c Given n∈Zand n&gt;2. We have to prove that for all m∈Z, n/m or n+m/nm. Suppose n/m and…

Q: 2. Prove that if m and n are integers and mn is even, then m is even or n is even.

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Q: Disprove the claim: 3 m E N such that m 2 3 and m? – 1 is a prime number. -

A: Given that ∃m∈ℕ such that m≥3 and m2-1 is a prime number. Take m=4. It is known that 4∈N And 4&gt;3.

Q: Give a direct proof of the claim: Vn E N, if the last two digits of n are divisible by 4 then n is…

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Q: Prove that for every n E N is a² | (a + 1)^+1. - 1)n+¹ na — (a + 1), a € N. - -

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Q: Let, a1 = 1, az = 2, b1 = 0, b2 = 1, for n 2 3, 2ап-1 + bn-1 for n 22 b, = bn–1+ an-1 an express an…

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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: 1. Given a deck of 30 cards labelled with numbers 1- 10 (with 3 cards of each number), how many…

A: From the given information,  A deck of 30 cards labelled with numbers 1-10 (with 3 cards of each…

Q: Given that a, =3 and if a is even; a n+1 5a -1 if a is odd. Then tern number four, i.e., a, is

A:

Q: Prove the following using the appropriate method.    Let a, b be integers. If ab is even, then at…

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Q: QUESTION 3 Prove that 2n > n2 if n is an integer greater than 4. For the toolhar press AIT. E10 /RCL

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Q: Suppose a, b, c and d are integers. Prove that  if a divides b and c divides d, then ac divides bd

A: According to the given information:   It is given that a divides b, there is an integer x such that…

Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: Show that if A and B are both n × n, then (a) det(AT BT ) = det(A) det(BT ); (b) det(AT BT ) =…

A: Given A and B are n×n matrix.

Q: Show that sup{ 1- n: n EN} = 1.

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Q: if m is even and n is odd demonstrate that m*n= is even

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Q: prove that, it ņ is an 3n+2is oald then integer and nis add

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