11. If n is an odd integer, show that n + 4n² + 11 is of the form 16k.

Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

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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: Determine for which positive integer values of n, 3n3 +2<n* and prove your claim by mathematical…

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Q: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

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Q: , 7/4 /213, 33 b) Solve e In(9) + 21 = 41 for æ. a) Write In(- -) in the form a lIn(7) + bln(3)…

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Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

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Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

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Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: 2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

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Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

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Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

Q: 3. Prove that if n is an odd integer then 7n+5 is even.

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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: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

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Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

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

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Q: Prove that 10 134n+1 + 115n+2 – 4 for all n E N.

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Q: For how many positive integers 'n', n3-8n2 + 20n - 13 is a prime?

A: Introduction :

Q: 1 ιπ 1s +472, e where n is an integer.

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Q: Prove that 12 - 2²+ 32 - ... .. + (-1)a-1n²=(-1)n-1n(n+1)/2

A: The given problem can be solved using mathematical induction:

Q: 3. Show that for all integers n: a) if d is an integer such that d |n + 8 and d | n² + 1, then d |…

A: To solve the given questions we will use the following formulas : 1.   We say that two positive…

Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: 3. By using mathematical induction proves 2' + 22 + 2 +.. +2" = 2"*! – 2 is true for all positive 3…

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Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: Prove n (2n = n n 1 k=1

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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: Use any method to deter (a) 13 The series 8.

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Q: 2. Prove that if n is even, then 3n + 1 is odd

A: The given expression is 3n + 1. Prove that 3n + 1 is odd, if n is even.

Q: Verify by mathematical induction Ž (2i – 1)° = n°(2n² – 1) for alln2 1. - i=1

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Q: Let n and k be integers with 1<k<n. Show that n n+1 Zk=1 (k - - n

A: Given Solution:

Q: an 34an-1 – 289an-2 for n > 3 with initial conditions aj = 68, az = -1734. an

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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: Prove that 11 is the only prime number of the form 4n2 - 25.

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Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

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

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Q: n. Prove that 21|(3n7 + 7n³ + 11n) for all integers

A:

Q: 2. Prove that if 2n² + 7n is even and n is an integer, then n is even.

A: We have to prove that the given statement is true by the help of mathematical induction

Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

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

Q: Prove that n^2 is an even number if and only if 3n + 1 is an odd number by proving  If n^2 is even…

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Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

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Q: Show that the sum of the first n positive odd integers, 1 + 3 + 5 + .......+ (2n - 1), is n2.

A: Given: 1+3+5+...........+2n-1 is n2

Q: If n is an odd integer, show that n4 + 4n2 + 11 is of the form 16k.

A: Solution: The objective is to show that given equation is of the form 16k

Q: give direct proof by contradiction that is 5n-1 is odd, then n is even

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Q: There is an integer n such that 7n2 – 5n + 11 is a prime number. -

A: Let fn=7n2-5n+11for n=1f1=7×1-5×1+11      =13 , which is prime number for n=2f2=722-5×2+11…